2024 Differential vs derivative

$The notion of the complex derivative is the basis of complex function theory. The definition of complex derivative is similar to the the derivative of a real function. However, despite a superficial similarity, complex differentiation is a deeply different theory. A complex function \ (f (z)\) is differentiable at a point \ (z_ {0}\in \mathbb .... Pokimane no makeup$

In calculus, the differential represents the principal part of the change in a function with respect to changes in the independent variable. The differential is defined by. where is …The derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function 's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear ... Jan 22, 2024 · Differential Vs. Derivative. In terms of relationships, the terms differential and derivative are closely related to one another. Variables are changing objects in mathematics, and the rate at which one variable changes in relation to another is known as a derivative. Dec 28, 2019 · Now, changing notation, we see that the total differential pops out as the action of the derivative on the vector (dx, dy): = (Δx, Δy) = (h, k), and so the image of the derivative is the equation of the tangent plane to f at the point (x0, y0), which provides an approximation to f itself in a presumably small neighborhood of (x0, y0)). Jan 23, 2024 · Read Differential and Derivative both are related but they are not the same. The main difference between differential and derivative is that a differential is an infinitesimal change in a variable, while a derivative is a measure of how much the function changes for its input. Note that if we are just given f (x) f ( x) then the differentials are df d f and dx d x and we compute them in the same manner. df = f ′(x)dx d f = f ′ ( x) d x. Let’s compute a couple of differentials. Example 1 Compute the differential for each of the following. y = t3 −4t2 +7t y = t 3 − 4 t 2 + 7 t.The names with respect to which the differentiation is to be done can also be given as a list of names. This format allows for the special case of differentiation with respect to no variables, in the form of an empty list, so the zeroth order derivative is handled through diff(f,[x$0]) = diff(f,[]).In this case, the result is simply the original expression, f.As nouns the difference between derivation and deviation. is that derivation is a leading or drawing off of water from a stream or source while deviation is the act of deviating; a wandering from the way; variation from the common way, from an established rule, etc.; departure, as from the right course or the path of duty.Jul 6, 2017 · 0. Let f: U ⊂ Rn → Rm be differentiable. The total derivative of f at a is the linear map dfa such that f(a + t) − f(a) = dfa(t) + o(t). For m = 1, the total differential of f is. df = m ∑ i = 1 ∂f ∂xidxi. Hope this helps. A partial derivative ( ∂f ∂t) of a multivariable function of several variables is its derivative with respect to one of those variables, with the others held constant. Let f(t, x) = t2 + tx + x2. Then ∂f ∂t = 2t + x + 0. On the other hand, the total derivative ( df dt) is taken with the assumption that all variables are allowed to vary.Calculus. #. This section covers how to do basic calculus tasks such as derivatives, integrals, limits, and series expansions in SymPy. If you are not familiar with the math of any part of this section, you may safely skip it. >>> from sympy import * >>> x, y, z = symbols('x y z') >>> init_printing(use_unicode=True)The comparison between differential vs. derivative is that the differential of a function is the actual change in the function, whereas the derivative is the rate at which the output value changes ...Apr 27, 2021 · Both gradient and total derivative are a collection or combination of the partial derivatives with respect to each input variable? Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their ... $\begingroup$ For example in my book the differential equation for the function "y=ax^2+bx+c" is d^3 y/dx^3=0 This equation contains the third order derivative of the variable "y" but the variable "y" itself is absent in this equation but yet the equation is considered as a differential equation according to the book which sounds against the …We have just seen how derivatives allow us to compare related quantities that are changing over time. In this section, ... Equation \ref{inteq} is known as the differential form of Equation \ref{diffeq}. Example $\PageIndex{4}$: Computing Differentials. For each of the following functions, find $dy$ and evaluate when $x=3$ and $dx=0.1.$Apr 25, 2016 · In particular, we can call the partial derivative $\frac{\partial f}{\partial x^k}(x)$, which will be a vector whose components are the partial derivatives of the components, following the above item. This goes on to a further ill-understanding of what dx d x actually means in integration. For example, if I have f(x) =x3 f ( x) = x 3, the derivative would be df(x) dx = 3x2 d f ( x) d x = 3 x 2. However, the differential, if I'm not mistaken, would be df(x) = 3x2dx d f ( x) = 3 x 2 d x. What is the difference between these two - the ...a qualitative or quantitative difference between similar or comparable things. Derivative Noun. Something derived. Differential Noun. (mathematics) an infinitesimal change in a variable, or the result of differentiation. Derivative Noun. (linguistics) A word that derives from another one. Differential Noun.Let dx, dy and dz represent changes in x, y and z, respectively. Where the partial derivatives fx, fy and fz exist, the total differential of w is. dz = fx(x, y, z)dx + fy(x, …Exercise 8.1.1 8.1. 1. Verify that y = 2e3x − 2x − 2 y = 2 e 3 x − 2 x − 2 is a solution to the differential equation y' − 3y = 6x + 4. y ′ − 3 y = 6 x + 4. Hint. It is convenient to define characteristics of differential …Mar 16, 2020 ... Comments ; Derivative Applications: Differentials - 05. Example. Sean Fitzpatrick · 117 views ; delta y vs. dy (differential). blackpenredpen · 262K&...It can refer to the difference between two values, rates of change, or the derivative of a function. In the context of mechanics, a differential is a device that allows the wheels of a vehicle to rotate at different speeds. This is necessary when turning, as the wheels on the inside of the turn need to rotate slower than the wheels on the ...The process of differentiation and integration are the two sides of the same coin. There is a fundamental relation between differentiation and integration. A...Discrete calculus or the calculus of discrete functions, is the mathematical study of incremental change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.The word calculus is a Latin word, meaning originally "small pebble"; as such pebbles were used for calculation, the meaning of the …No, no and no: they are very different things. The derivative (also called differential) is the best linear approximation at a point. The directional derivative is a one-dimensional object that describes the "infinitesimal" variation of a function at a point only along a prescribed direction. I will not write down the definitions here. Explain the relationship between differentiation and integration. In the previous two sections, we looked at the definite integral and its relationship to the area under the curve of a function. Unfortunately, so far, the only tools we have available to calculate the value of a definite integral are geometric area formulas and limits of Riemann ...Limits and derivatives are extremely crucial concepts in Maths whose application is not only limited to Maths but are also present in other subjects like physics. In this article, the complete concepts of limits and derivatives along with their properties, and formulas are discussed. This concept is widely explained in the class 11 syllabus.The estimate for the partial derivative corresponds to the slope of the secant line passing through the points (√5, 0, g(√5, 0)) and (2√2, 0, g(2√2, 0)). It represents an approximation to the slope of the tangent line to the surface through the point (√5, 0, g(√5, 0)), which is parallel to the x -axis. Exercise 13.3.3.The derivative of the difference of a function $f$ and a function $g$ is the same as the difference of the derivative of $f$ and the derivative of $g$. The derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function. In simple words, directional derivative can be visualized as slope of the function at the given point along a particular direction. For example partial derivative w.r.t x of a function can also be written as directional derivative of that function along x direction.Calculus has two main parts: differential calculus and integral calculus. Differential calculus studies the derivative and integral calculus studies (surprise!) the integral. The derivative and integral are linked in that they are both defined via the concept of the limit: they are inverse operations of each other (a fact sometimes known as the ...A differential is a small change in a variable, while a derivative is the rate of change of a function at a specific point. For example, if we have a function f (x) = x^2, the differential of f (x) with respect to x is dx, while the derivative of f (x) at x = 2 is 4. Discrete differential calculus is the study of the definition, properties, and applications of the difference quotient of a function. The process of finding the difference quotient is called differentiation. Given a function defined at several points of the real line, the difference quotient at that point is a way of encoding the small-scale (i ...Differentiation and Integration are the two major concepts of calculus. Differentiation is used to study the small change of a quantity with respect to unit change of another. (Check the Differentiation Rules here). On the other hand, integration is used to add small and discrete data, which cannot be added singularly and representing in a ...In Willie Wong's reply to one question, he used some concepts: "interior derivative" of a differential form and "exterior derivative" of a scalar function on $\mathbb{R}^3$. For "exterior derivative" of a scalar function on $\mathbb{R}^3$, I think it means the exterior derivative of the scalar function viewed as a 0-form. Jun 15, 2019 ... ... differentiation and integration 4:31 integral of the derivative of the function 5:18 Fundamental theorem of Calculus 7:12 anti-derivative or ...The derivative of a function (if it exists) is just another function. Saying that a function is differentiable just means that the derivative exists, while saying that a function has a continuous derivative means that it is differentiable, and its …Integration is a method to find definite and indefinite integrals. The integration of a function f (x) is given by F (x) and it is represented by: where. R.H.S. of the equation indicates integral of f (x) with respect to x. F (x) is called anti-derivative or primitive. f (x) is called the integrand. dx is called the integrating agent.Difference Rule. The derivative of the difference of a function $f$ and a function $g$ is the same as the difference of the derivative of $f$ and the derivative of $g$ : …is an ordinary differential equation since it does not contain partial derivatives. While. ∂y ∂t + x∂y ∂x = x + t x − t (2.2.2) (2.2.2) ∂ y ∂ t + x ∂ y ∂ x = x + t x − t. is a partial differential equation, since y y is a function of the two variables x x and t t and partial derivatives are present. In this course we will ...A partial derivative ( ∂f ∂t) of a multivariable function of several variables is its derivative with respect to one of those variables, with the others held constant. Let f(t, x) = t2 + tx + x2. Then ∂f ∂t = 2t + x + 0. On the other hand, the total derivative ( df dt) is taken with the assumption that all variables are allowed to vary.Key Difference: In calculus, differentiation is the process by which rate of change of a curve is determined. Integration is just the opposite of differentiation. It sums up all small area lying under a curve and finds out the total area. Differentiation and Integration are two building blocks of calculus. Differential calculus and Integral ...Differentiation Noun. a discrimination between things as different and distinct; ‘it is necessary to make a distinction between love and infatuation’; Derivative Noun. (calculus) The derived function of a function (the slope at a certain point on some curve f (x)) ‘The derivative of f:f (x) = x^2 is f’:f' (x) = 2x ’; Differentiation Noun.$\begingroup$ For example in my book the differential equation for the function "y=ax^2+bx+c" is d^3 y/dx^3=0 This equation contains the third order derivative of the variable "y" but the variable "y" itself is absent in this equation but yet the equation is considered as a differential equation according to the book which sounds against the …Always thinking the worst and generally being pessimistic may be a common by-product of bipolar disorder. Listen to this episode of Inside Mental Health podcast. Pessimism can feel...So, first get the formula for the differential. \[dV = 4\pi {r^2}dr\] Now compute $dV$. \[\Delta V \approx dV = 4\pi {\left( {45} \right)^2}\left( {0.01} \right) = …The Gateaux differential generalizes the idea of a directional derivative. Deﬁnition 1. Let f : V !U be a function and let h 6= 0 and x be vectors in V. The Gateaux differential d h f is deﬁned d h f = lim e!0 f(x +eh) f(x) e. Some things to notice about the Gateaux differential: There is not a single Gateaux differential at each point. Calculus. #. This section covers how to do basic calculus tasks such as derivatives, integrals, limits, and series expansions in SymPy. If you are not familiar with the math of any part of this section, you may safely skip it. >>> from sympy import * >>> x, y, z = symbols('x y z') >>> init_printing(use_unicode=True)Differentiation is a related term of different. As nouns the difference between different and differentiation is that different is the different ideal while differentiation is the act of differentiating. As an adjective different is not the same; exhibiting a difference.The mathematical depiction of derivatives and gradients as vector or scalar figures is a further significant difference between them. Derivatives are scalar values that represent just one value that represents how quickly a function changes. They give details about a tangent line’s slope to the curve at some point. The derivative essentially ...The reason is because for a function the be differentiable at a certain point, then the left and right hand limits approaching that MUST be equal (to make the limit exist). For the absolute value function it's defined as: y = x when x >= 0. y = -x when x < 0. So obviously the left hand limit is -1 (as x -> 0), the right hand limit is 1 (as x ... To get a quick sale, it is essential to differentiate your home from others on the market. But you don't have to break the bank to improve your home's… In order to get a quick sale...Free derivative calculator - differentiate functions with all the steps. Type in any function derivative to get the solution, steps and graph So what does ddx x 2 = 2x mean?. It means that, for the function x 2, the slope or "rate of change" at any point is 2x.. So when x=2 the slope is 2x = 4, as shown here:. Or when x=5 the slope is 2x = 10, and so on. A derivative basically finds the slope of a function. In the previous example we took this: h = 3 + 14t − 5t 2. and came up with this derivative: d dt h = 0 + 14 − 5 (2t) = 14 − 10t. Which tells us the slope of the function at any time t. We used these Derivative Rules: The slope of a constant value (like 3) is 0.In differential calculus, there is no single uniform notation for differentiation.Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context.If we take the derivative of a function y=f(x), the unit becomes y unit/x unit. A derivative is the tangent line's slope, which is y/x. So the unit of the differentiated function will be the quotient. For example, v(t) is the derivative of s(t). s -> position -> unit: meter t -> time -> unit: second Nov 10, 2020 · the differential $dx$ is an independent variable that can be assigned any nonzero real number; the differential $dy$ is defined to be $dy=f'(x)\,dx$ differential form given a differentiable function $y=f'(x),$ the equation $dy=f'(x)\,dx$ is the differential form of the derivative of $y$ with respect to $x$ Differential of a function. In calculus, the differential represents the principal part of the change in a function with respect to changes in the independent variable. The differential is defined by. where is the derivative of f with respect to , and is an additional real variable (so that is a function of and ).Solving a differential equation means finding the value of the dependent variable in terms of the independent variable. The following examples use y as the dependent variable, so the goal in each problem is to solve for y in terms of x. An ordinary differential equation (ODE) has only derivatives of one variable — that is, it has no …Learn about derivatives as the instantaneous rate of change and the slope of the tangent line. This video introduces key concepts, including the difference between average and instantaneous rates of change, and how derivatives are central to differential calculus. The derivative of the difference of a function $f$ and a function $g$ is the same as the difference of the derivative of $f$ and the derivative of $g$. The derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function. Differentiation is a method of finding the derivative of a function. Differentiation is a process, in Maths, where we find the instantaneous rate of change in function based on …The mathematical depiction of derivatives and gradients as vector or scalar figures is a further significant difference between them. Derivatives are scalar values that represent just one value that represents how quickly a function changes. They give details about a tangent line’s slope to the curve at some point. The derivative essentially ...Your friend is wrong, or you misinterpreted him. You can differentiate functions fine, what you friend probably meant are tensor fields (or in general, sections of non-trivial vector bundles). Now to show the connection to differential forms, I want to say something about what $ \mathrm d ^ 2 x $, $ \mathrm d x ^ 2 $, and so forth really mean.As you probably know, one way to think of an exterior differential form is as a multilinear alternating (or antisymmetric) operation on tangent vectors.Key Difference: In calculus, differentiation is the process by which rate of change of a curve is determined. Integration is just the opposite of differentiation. It sums up all small area lying under a curve and finds out the total area. Differentiation and Integration are two building blocks of calculus. Differential calculus and Integral ...1. @Soeren I think symbolic diff normally gives you an entire equation expression, while autodiff only evaluates the basic differentiation rules without requiring a final equation. For example, (x1 * x2 * sin (x3) - exp (x1 * x2)) / x3, the symbolic diff will return the grad expression w.r.t x1, x2 and x3 separately.Hence, any covariant derivative would yield the very same result. ∇uv ∇ u v is a very different object. It is also a vector, so it is convenient for us to write it acting on a function f f to compare with the previous expression. In components, we have. (∇uv)μ = uν∇νvμ, = uν ∂ ∂xνvμ +uνΓμνρvρ. ( ∇ u v) μ = u ν ∇ ...VANCOUVER, British Columbia, Dec. 23, 2020 (GLOBE NEWSWIRE) -- Christina Lake Cannabis Corp. (the “Company” or “CLC” or “Christina Lake Cannabis... VANCOUVER, British Columbia, D...Explain the relationship between differentiation and integration. In the previous two sections, we looked at the definite integral and its relationship to the area under the curve of a function. Unfortunately, so far, the only tools we have available to calculate the value of a definite integral are geometric area formulas and limits of Riemann ...Nov 17, 2020 · 1 Answer. Sorted by: 1. In simplistic terms, a differential relates to the increase in the value of a function, an object taking a scalar as argument and returning a scalar, for a "small" variation in the independent variable. A variation relates to the increase in the value of a functional, and object taking a function as argument and ... Discrete calculus or the calculus of discrete functions, is the mathematical study of incremental change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.The word calculus is a Latin word, meaning originally "small pebble"; as such pebbles were used for calculation, the meaning of the …It is all about slope! Slope = Change in Y Change in X Let us Find a Derivative! To find the derivative of a function y = f (x) we use the slope formula: Slope = Change in Y Change …Free derivative calculator - differentiate functions with all the steps. Type in any function derivative to get the solution, steps and graph In mathematics, the partial derivative of any function having several variables is its derivative with respect to one of those variables where the others are held constant. The partial derivative of a function f with respect to the differently x is variously denoted by f’ x ,f x, ∂ x f or ∂f/∂x. Here ∂ is the symbol of the partial ... The reason is because for a function the be differentiable at a certain point, then the left and right hand limits approaching that MUST be equal (to make the limit exist). For the absolute value function it's defined as: y = x when x >= 0. y = -x when x < 0. So obviously the left hand limit is -1 (as x -> 0), the right hand limit is 1 (as x ... The derivative of the function secant squared of x is d/dx(sec^2(x)) = 2sec^2(x)tan(x). This derivative is obtained by applying the chain rule of differentiation and simplifying th...It breaks the term ‘ adaptive teaching’ into more concrete recommendations for teaching. For example: Adapting lessons, whilst maintaining high expectations for all, so that all pupils have the opportunity to meet expectations. Balancing input of new content so that pupils master important concepts. Making effective use of teaching assistants.Taking the derivative at a single point, which is done in the first problem, is a different matter entirely. In the video, we're looking at the slope/derivative of f (x) at x=5. If f (x) were horizontal, than the derivative would be zero. Since it isn't, that indicates that we have a nonzero derivative. Show more...Similarly, here's how the partial derivative with respect to y ‍ looks: ∂ f ∂ y ( x 0, y 0, …) = lim h → 0 f ( x 0, y + h, …) − f ( x 0, y 0, …) h ‍. The point is that h ‍ , which represents a tiny tweak to the input, is added to different input variables depending on which partial derivative we are taking.VANCOUVER, British Columbia, Dec. 23, 2020 (GLOBE NEWSWIRE) -- Christina Lake Cannabis Corp. (the “Company” or “CLC” or “Christina Lake Cannabis... VANCOUVER, British Columbia, D...A Directional Derivative is a value which represents a rate of change; A Gradient is an angle/vector which points to the direction of the steepest ascent of a curve. Let us take a look at the plot of the following function: $$ \bbox[lightgray] {f(x) = -x^2+4}\qquad (1)$$In Willie Wong's reply to one question, he used some concepts: "interior derivative" of a differential form and "exterior derivative" of a scalar function on $\mathbb{R}^3$. For "exterior derivative" of a scalar function on $\mathbb{R}^3$, I think it means the exterior derivative of the scalar function viewed as a 0-form.

A function is differentiable at an interior point a of its domain if and only if it is semi-differentiable at a and the left derivative is equal to the right derivative. An example of a semi-differentiable function, which is not differentiable, is the absolute value function. f ( x ) = | x | {\displaystyle f (x)=|x|} , at a = 0. We find easily.. Cinderella movies cartoon

derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined differentiable at $a$ a function for …Differential Calculus is a branch of Calculus in mathematics that deals with the study of the rates at which quantities change. It involves calculating derivatives and …An ordinary differential equation involves a derivative over a single variable, usually in an univariate context, whereas a partial differential equation involves several (partial) derivatives over several variables, in a multivariate context. E.g. $$\frac{dz(x)}{dx}=z(x)$$ vs.Fréchet derivative. In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used ... Derivatives of sin (x), cos (x), tan (x), eˣ & ln (x) Derivative of logₐx (for any positive base a≠1) Worked example: Derivative of log₄ (x²+x) using the chain rule. Differentiating logarithmic functions using log properties.Nov 16, 2022 · Note that if we are just given f (x) f ( x) then the differentials are df d f and dx d x and we compute them in the same manner. df = f ′(x)dx d f = f ′ ( x) d x. Let’s compute a couple of differentials. Example 1 Compute the differential for each of the following. y = t3 −4t2 +7t y = t 3 − 4 t 2 + 7 t. That is the definition of the derivative. So this is the more standard definition of a derivative. It would give you your derivative as a function of x. And then you can then input your particular value of x. Or you could use the alternate form of the derivative. If you know that, hey, look, I'm just looking to find the derivative exactly at a.There are three things we could talk about. The derivative/differential, the partial derivative (w.r.t a particular coordinate) and the total derivative (w.r.t a particular coordinate). The definition of the first varies, but the definitions all …In the section we introduce the concept of directional derivatives. With directional derivatives we can now ask how a function is changing if we allow all the independent variables to change rather than holding all but one constant as we had to do with partial derivatives. In addition, we will define the gradient vector to help with some …More formally, we make the following definition. Definition 1.7. A function f f is continuous at x = a x = a provided that. (a) f f has a limit as x → a x → a, (b) f f is defined at x = a x = a, and. (c) limx→a f(x) = f(a). lim x → a f ( x) = f ( a). Conditions (a) and (b) are technically contained implicitly in (c), but we state them ...This goes on to a further ill-understanding of what dx d x actually means in integration. For example, if I have f(x) =x3 f ( x) = x 3, the derivative would be df(x) dx = 3x2 d f ( x) d x = 3 x 2. However, the differential, if I'm not mistaken, would be df(x) = 3x2dx d f ( x) = 3 x 2 d x. What is the difference between these two - the ...Nov 10, 2020 · the differential $dx$ is an independent variable that can be assigned any nonzero real number; the differential $dy$ is defined to be $dy=f'(x)\,dx$ differential form given a differentiable function $y=f'(x),$ the equation $dy=f'(x)\,dx$ is the differential form of the derivative of $y$ with respect to $x$ The symbol Δ Δ refers to a finite variation or change of a quantity – by finite, I mean one that is not infinitely small. The symbols d, δ d, δ refer to infinitesimal variations or numerators and denominators of derivatives. The difference between d d and δ δ is that dX d X is only used if X X without the d d is an actual quantity that ...Differential vs Derivative: Comparison Chart. Ringkasan Diferensial Vs. Turunan. Dalam matematika, laju perubahan satu variabel terhadap variabel lain disebut turunan dan persamaan yang menyatakan hubungan antara variabel-variabel ini dan turunannya disebut persamaan diferensial. Integral calculus was one of the greatest discoveries of Newton and Leibniz. Their work independently led to the proof, and recognition of the importance of the fundamental theorem of calculus, which linked integrals to derivatives. With the discovery of integrals, areas and volumes could thereafter be studied. Integral calculus is the second …We can also define a derivative in terms of differentials as the ratio of differentials of function by the differential of a variable. A derivative is the change in a function ($\frac{dy}{dx}$); a differential is the change in a variable $ (dx)$. A function is a relationship between two …1. @Soeren I think symbolic diff normally gives you an entire equation expression, while autodiff only evaluates the basic differentiation rules without requiring a final equation. For example, (x1 * x2 * sin (x3) - exp (x1 * x2)) / x3, the symbolic diff will return the grad expression w.r.t x1, x2 and x3 separately.That is the definition of the derivative. So this is the more standard definition of a derivative. It would give you your derivative as a function of x. And then you can then input your particular value of x. Or you could use the alternate form of the derivative. If you know that, hey, look, I'm just looking to find the derivative exactly at a..

Popular Topics