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Questions tagged [derivatives]

Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value)

Derivative of a function has a very natural geometric and physical interpretation: it corresponds to slope of the tangent line and to instantaneous velocity. In applications, it usually describes the rate of change of a physical variable.

Basic techniques used for computing the derivative of a given function are

It is useful to know the derivatives of elementary functions. This tag is intended for questions on the evaluation of derivatives.

Derivatives may be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.

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Proving $f$ is linear with $\dfrac{f(b)-f(a)}{b-a}=\sup\left\{f^{'}(x) :x\in[a,b]\right\}$

Let $f\in C^1([a,b],\mathbb{R})$ such as $\dfrac{f(b)-f(a)}{b-a}=\sup\left\{f^{'}(x) : x\in[a,b]\right\}$. Prove that $f$ is linear Proving that $f$ is linear means to prove that $\forall x\in [a,b],\, f^{'}(x) = \dfrac{f(b)-f(a)}{b-a}$. Because…
Sewshley
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Odd definition of derivative of a function

I saw a bit different definition of derivative fiven by $f(x+\epsilon) = f(x)+f'(x)\epsilon + {\cal O}(\epsilon^2)$, where the ${\cal O}(\epsilon^2)$ stands for any function $g$ which satisfies that for all $x$ sufficiently large exists $c>0$ in…
piero
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Difficulty understanding partial derivatives

If I have a function: $z=f(x, y)$ And I want to find: $(\frac{ \partial y}{ \partial x}) _{z}$ How do I do this? My approach is that if z is kept constant, $dz$ would be $0$. Therefore, using the relation: $dz=( \frac{ \partial z}{ \partial x})dx +…
ED2468
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When is a composite function differentiable at a point.

We all know that if a function g is differentiable at a point a, and f is differentiable at g(a), then f∘g is differentiable at a with derivative f'(g(a))g'(a). However, it is not necessary for both of them to be differentiable at their respective…
Aqeel
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Derivative of a quadratic form with respect to matrix

I've got a silly question, but I don't see, where I'm wrong. Given $x,y \in \mathbb{R}^D$ and $A$ is a symmetric matrix. $$ f(A) = (x-y)^T A (x-y)\\ \frac{\partial{f}}{\partial{A}} = (x-y)(x-y)^T $$ But if I expand brackets in $f(A)=x^T A x - 2x^T A…
Joseph Kirtman
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Why are the graphs of two initial functions that arise from the same equation different?

So I have f'(x)=-f(x)<=>f'(x)+f(x)=0, let h(x)=f'(x)+f(x)=0, I found two ways to approach this: e^x•f'(x)+e^x•f(x)=0<=>(e^x•f(x))'=0 (f'(x)/f(x))+1=0<=>(ln(|f(x)|)+x)'=0 So h(x) is either 1) or 2), but I found that the 2) one is equal to…
Nick The Greek
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Regarding replacement of a variable with a constant in rate of change.

Let's consider a circle whose area is increasing at a constant rate of 5m²/s.We are to find the rate of change of radius of the circle at the instant when the radius of the circle is 2.5cm. This is how I proceeded. *Let area of the circle at time t…
Arghya Deb
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derivative of multinomial

Can you please show me how to find the second derivative by using the general formula? The general form will follow a multinomial pattern. Let u,v,w be functions of x. Then: $$\frac{d^n}{dx^n}(u\cdot v\cdot…
Tony
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Confusing derivative result

I've been experimenting with some differentiation, specifically the $x^x$ function. Using a table of standard derivatives, $\frac{d}{dx}x^n $ should be equal to $nx^{n-1}$. Using this logic, if $n = x$, then $\frac{d}{dx}x^x $ should be equal to $x…
Mark Smith
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Implicit functions and derivatives

Suppose we want to find the roots of a function $f$ which depends on $x$,$y$ and $r$, i.e. we are solving $f(x,y,r) = 0$. If $r(x,y)$ is a solution to this equation (there doesn't have to be an analytic expression for $r(x,y)$ a priori), can we then…
Geigercounter
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Example of infinitely differentiable function such that $f^{(k)}(0) = a_k$

I want to find an infinitely differentiable function such that $f^{(k)}(0) = a_k$, where $a_k$ is some sequence of real numbers. My attempt was to raise $e$ to some power, but I have not gotten far. I feel like there should be a pretty simple simple…
ABlack
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Why does $\ \frac{x (\ln a)' \ln x}{\ln a} = 0 $?

From "some proof that uses the definition" of a derivative, I understand that: $$\ (\log_a x)' = \frac{1}{x \ln a} $$ Since this is the case, I should also be able to prove that: $$\ \left(\frac{\ln a}{\ln a}\right)' = \frac{1}{x \ln a} $$ So I did…
jayphur
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Suppose f(x,y) is a function of two independent variables x and y. Is dy/dx always zero?

I am confused about the derivative of one independent variable with respect to another independent variable. Is it always zero or is it always undefined?
Ravi
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Using dot notation, what is the derivative of $r^2$ with respect to t, when r is a function of t?

I'm a bit confused about using dot notation; I have $A = r^2$, and I need the derivative. Would this be $\frac{dA}{dt}=\dot{r}^2$, $\frac{dA}{dt}=2\dot{r}$, or simply $\frac{dA}{dt}=2r$? Thanks for any help!
Barney_Dinosaur
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Is my calculation of third order derivative correct?

I need to find the third order derivative of $\frac{1}{s+3}$. I know that I need to calculate the derivative of the derivative till I get to the desired order. So, here are my answers: First-order derivative: $\frac{-1}{(s+3)^2}$ Second-order…
An SO User
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