Mathematics Stack Exchange
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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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Derivative and limit of $\sqrt[3]{x-\sin x}$

Given $f:R\rightarrow R, f(x) = \sqrt[3]{x-\sin x}$, compute $f'(0)$.Now, this can be done by using the definition of the derivative : $$f'(x_0) = \lim_{x\to{x_0}} \frac{f(x)-f(x_0)}{x-x_0}$$This yields the right answer.However, why can't we derive…
NotADeveloper
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How to simplify derivatives

The math problem asks to find the derivative of the function $$y=(x+1)^4(x+5)^2$$ I get to the part $$(x+1)^4 \cdot 2(x+5) + (x+5)^2 \cdot 4(x+1)^3$$ How do they arrive at the answer $$2(x+1)^3(x+5)(3x+11) ?$$
Fiona Lu
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How come $\frac{1}{1-x}$ and $\frac{x}{1-x}$ have the same derivative?

Maybe I'm just having a brain breakdown moment, but it seems weird to me that both functions have exactly the same derivative, namely $\frac{1}{(1-x)^2}$. Obviously I'm not disputing whether or not it's correct, but I'm looking for some sort of…
jeremy radcliff
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Is a function strictly increasing if its derivative is positive at all point but critical points?

$f: (a,b) \to \Bbb{R}$ is differentiable and $f'(x)>0$ at all points but at $c$ where $f'(c) = 0$. I need to prove that $f$ is strictly increasing. I thought to split the intervals to $(a,c)$ and $(c,b)$ and use the continuity of $f$ at $c$, but I'm…
Itay4
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The derivative of $\arccos(\cos(x))$

I was asked to show that $\frac{d}{dx}\arccos(\cos{x}), x \in R$ is equal to $\frac{\sin{x}}{|\sin{x}|}$. What I was able to show is the following: $\frac{d}{dx}\arccos(\cos(x)) = \frac{\sin(x)}{\sqrt{1 - \cos^2{x}}}$ What justifies equating…
Kariem
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How to differentiate the square root of the function inside another square root of the function?

If a function is defined as sum of radicals in another radicals. Then how to differentiate this function $$\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\dots}}}\quad?$$
Zonnie
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I am having trouble finding the first derivative of $R(P) = (Pe^r)(1-\frac{P}{K})$

I am having trouble finding the first derivative of $R(P) = (P\textrm{e}^r)(1-\frac{P}{K})$ I am told to use the product rule for this. The first part of it $(P\textrm{e}^r)$ I believe will remain the same for the derivative of it. I am struggling…
Amanda Smith
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quant interview question from 1996 at Banc One in Columbus Ohio

During a quant interview with Banc One in 1996 post-physics doctorate, I choked on this interview question: What is the derivative $\frac{dy}{dx}$ of $y=x^{x^{x^{.^{.^{.}}}}}$
phdmba7of12
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Prove $\frac{d^2y}{dx^2}=-\frac{\frac{d^2x}{dy^2}}{(\frac{dx}{dy})^3}$

Prove…
Matata
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Is there a function $f(x)$ which is defined near $x = c$ and infinitely differentiable near $x = c$ and satisfy the following properties

Is there a function $f(x)$ which is defined near $x = c$ and infinitely differentiable near $x = c$ and satisfy the following properties: For any positive real number $\delta$, there exist real numbers $x, x^{'}$ such that $c - \delta < x, x^{'} <…
Bolt
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Why $\frac{d}{dt}f(x+t(y−x))<0$ if $x < y, f(y) < f(x)$

Here excerpt from a book: Аssume that $f$ satisfies $\nabla f(x) \ge 0$ for all $x$, but is not nondecreasing, i.e., there exist $x,y$ with $x < y$ and $f(y) < f(x)$. By differentiability of $f$ there exists at $t\in[0,1]$…
Yola
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Why is $\frac{d(x^n)}{d(x)}=nx^{n-1}$

So I was thinking about what I have learnt and I realised that I kind of took the derivative of a function for granted. So I did some research as I wanted to find out how this was discovered and I stumbled upon this. More specially, here is a…
YellowPillow
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Is the $x$-axis a differentiable function?

Is the $x$-axis a differentiable function?
taniya kapoor
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Differentiate $\sqrt{1+e^x}$ using the definition of a derivative

This is the progress I've made so far. $$\lim_{h \to 0} \frac{\sqrt{1+e^{x+h}}-\sqrt{1+e^{x}}}{h}$$ $$= \lim_{h \to 0} \frac{\left(1+e^{x+h}\right)-\left(1+e^{x}\right)}{h\left(\sqrt{1+e^{x+h}}+\sqrt{1+e^{x}}\right)}$$ $$= \lim_{h \to 0}…
Mr. Bump
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differentiation problem

If $x^{13}y^{7}=(x+y)^{20}$ , then $\frac{dy}{dx}$ directly doing it makes it very complicated so, I did this $\left(\frac{x}{y}\right)^{13}=\left(1+\frac{x}{y} \right)^{20}$. following are the options for solution (a) $\frac{y^2}{x^2}$…
Onix
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