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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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Step-by-step derivative of $\left ( \frac{c_1 x}{c_2 x + c_3 + c_4 \sqrt{c_5 x}} \right)^{c_6x + c_7 + c_8 \sqrt{c_9 x}}$

Can someone please walk step by step on how to calculate the derivative $\left ( \frac{c_1 x}{c_2 x + c_3 + c_4 \sqrt{c_5 x}} \right)^{c_6x + c_7 + c_8 \sqrt{c_9 x}}$ Where the $c_i$ are constants, which are positive.
user341502
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How to show that this function is differentiable in $x=0$ but not in $x=1$

I am stuck with showing that this function is differentiable at $x=0$ but not at $x=1$. $f(x) = \begin{cases} x^2, & \text{if x}\in{\mathbb{Q}}\\ x^3, & \text{if x}\notin{\mathbb{Q}}\\ \end{cases}$ So for $x=0$: Case 1: $h \in…
ffritz
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Find stationary point of $y = \frac {e^{2x}} {4 + e^{3x}}$

The curve with equation $y = \frac {e^{2x}} {4 + e^{3x}}$ has one stationary point. Find the exact values of the coordinates of this point. I got to the point where this is my $\frac {dy} {dx}$:$$\frac{ (4 + e^{3x})…
Riduan Gonzalez
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Need to prove $f(x)=0$ for all $x \in R$

$f(x)$ is infinitely differentiable and $∃ L∈\mathbb{R}$ such that $|f^{(k)}(x)|≤L$ for any $k∈\mathbb{N}$. I need to prove that : If $f(1/n)=0$ then $f(x)=0$ for any $x∈\mathbb{R}$.
Barna Sarkar
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Differentiation of a function

We need to find out the derivative $\frac{dy}{dx}$ of the following: $x^m$$y^n$=$(x+y)^{m+n}$ I know how to differentiate the function and on solving we get $\frac{dy}{dx}$=$\frac{y}{x}$. But we notice that $\frac{dy}{dx}$ is independent of values…
Piyush Raut
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About derivative of a function

I know the limit definition of a derivative. And I read at the end that if the limit exist then we say that function is differentiable and this existing limit is denoted by $\dfrac{d}{dx}$ . so we say that $\dfrac{d}{dx}$ is a notation. But…
saman kumar
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How do I take the derivative of y as well as x?

$$\frac{\mathrm{d}(y^3-4x)}{\mathrm{d}x}=\frac{\mathrm{d}2}{\mathrm{d}x}$$ Note that $\text{RHS}$ (Right Hand Side) is $0$, since $2$ is constant. By the linearity of the differential operator, we can…
Byun Baekjin
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Implicit differentiation from Larson 13.5

Implicit differentiation: $$\frac{x}{x^{2}+y^{2}}-y^{2}=5$$ I've tried several ways including Wolfram, and the answer is not getting accepted. This is how I did it so far:
Jenny Kay
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Bernstein polynomial derivative

I am having trouble visualizing where the negative sign comes from in the final answer when differentiating the Bernstein polynomial: $$\frac{\mathrm d}{\mathrm du}\left[B_{i,n}(u)\right] = \frac{\mathrm d}{\mathrm…
c0der
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Name for functions whose derivative is expressed in terms of the function value.

Is there a name for the class of functions whose derivative is expressed only in terms of the function value? One example is the exponential, another example is \begin{align} s_1(t) = \frac{1}{1 + e^{-t}} \end{align} with…
Escualo
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Exercise on a fixed end Lagrange's MVT

Given a function f with derivative on all $[a;b]$ with $f'(a) = f'(b)$, show that there exists $c \in (a;b)$ such that $f'(c) = \frac{f(c)-f(a)}{c-a}$. This is some kind of MVT with constraint. I have a proof but it uses Darboux's theorem. Can you…
deufeufeu
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Differentiating $y=x^6e^{-4x^3}$

Can some one explain, how to solve this derivative. I'm total beginner. Would be preferable if some one explain step-by-step. $$y=x^6e^{-4x^3}$$
Viktor
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$f (x, y)$ is given by $f (x, y) = (x^2 - 5x\cdot y)\cdot e^y$

These are the question to that function that I'm struggling with: Find the partial derivatives of first and second order of $f(x, y)$. Find the stationary points of $f(x, y)$ and determines for each point on it/they are a local maximum point, the…
Frank Len
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Given that $\cos(x/2)\cos(x/4)\cos(x/8)\ldots=(\sin x)/x,$ prove that $(1/2^2)\sec^2(x/2)+(1/2^4)\sec^2(x/4)\ldots=\csc^2(x) - (1/x^2).$

How to solve this one I know this is related to differentiation but how to proceed with this??? Please give all steps so that it is easily understood.
Rahul Laddha
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Find acceleration given velocity with respect to distance

A particle moves with the velocity given by: $$v(s(t)) = \frac{3s(t) + 4}{2s(t)+1}$$ where s(t) is the distance traveled. Find the acceleration when s(t) = 2. My attempt: $$a(s(t)) = \frac{\mathrm d}{\mathrm dV(s(t))}\left(\frac{3s(t) +…
I_snort_TNT
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