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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Nth derivative can be expressed like that?

$$ \frac{f^{(n)}(z_0)}{n!} = \lim_{z \rightarrow z_0} \frac{f(z) -f(z_0)}{(z-z_0)^n} $$ Why is that nth derivative can be expressed like that limit of quotient? I can understand the meaning but I couldnt get closed form equation. Thanks.
HCCHUNG
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The basic of functional derivative

I've just started to learn mathematical physics, and I read Stone and Goldbart's Mathematics for Physics. But right at the beginning when they introduce the functional derivative, I couldn't understand their explanation: 1.2.1. The functional…
Kim Dong
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Any derivatives that are difficult to take?

Many integrals are either very complicated or in some cases impossible? However, I have not yet seen derivatives that are very difficult or cannot be solved for. Does anyone have an example of a derivative that cannot be solved through standard…
mtheorylord
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Does the derivative exist?

So I have this question here: $f(x)=\frac{1}{1+|x|}+\frac{1}{1+|x-a|},a>0$ I am asked to find the derivative and I correctly found it as: $f'(x)=-\frac{x-a}{|x-a|(|x-a|+1)^2}-\frac{x}{|x|(|x|+1)^2},a>0$ I am then asked to determine if $f'(0)$ and…
Future Math person
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Find the $n$th derivative of $f(x)=(\sqrt{x^2-1}+\sqrt{x-1})^2$

In this case $n=16$ and the point is $x=1$. I know $\sqrt{x}$ is not differentiable in zero, but this function actually has left derivatives. Also, I know that $f(x)=(\sqrt{x^2-1}+\sqrt{x-1})^2=x^2+x-2+2\sqrt{(x^2-1)(x-1)}$, so it is enogth to…
Luis
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Can we use this as definition of the derivative?

Let $D$ be a subset of $\mathbb{K}$ where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. Let $f:D\to \mathbb{K}^N$ be a function. Then $f$ is differentiable at $x$ if $$\lim_{n\to \infty}\frac{f(x+\frac{1}{n})-f(x)}{\frac{1}{n}}$$ exists.
satokun
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Higher order derivatives of the binomial factor

Let $p$,$l$ be positive integers and $\theta$ be a parameter. The question is to compute the following quantity: \begin{equation} \kappa^{(p)}_l := \left. \frac{\partial^p}{\partial \theta^p} \binom{\theta}{l} \right|_{\theta=0} \end{equation} With…
Przemo
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Proof of $\frac{d \sqrt{x}}{dx}$ including proof of the limit?

Looking at proofs for $\frac{d \sqrt{x}}{dx}$, ($0
James Snook
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Can the endpoints of the interval considered satisfy the mean value theorem?

For example, if you have a graph $y=x$ and you want to find the values of $c$ that satisfy the mean value theorem for $x\in[1, 3]$, do the points $c=1$ and $c=3$ count as valid? I only ask because for a homework problem $y=-x^3+4x^2-3; [0, 4]$ find…
Jack Pan
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Partial derivatives chain rule

Let $ w=\frac{x^2}{y} $ with $ x=e^{-u^2}u $ and $ y=e^{-u^2}v $ $ \frac{\partial w}{\partial x} = (x^2)'\frac{1}{y} = \frac{2x}{y} $ $ \frac{\partial w}{\partial y} = x^2(\frac{1}{y})' = -\frac{x^2}{y^2} $ $ \frac{\partial y}{\partial u} =…
Edward Ruchevits
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Derivative of $\frac{\ln{x}}{\ln{a}}$

I'm asked to prove that the derivative of $$\frac{\ln{x}}{\ln{a}}$$ is $$\frac{1}{x\cdot\ln{a}}$$ My attempt: $$\frac{d}{dx}(\frac{\ln{x}}{\ln{a}}) = \frac{\frac{1}{x}\cdot \ln{a} - \frac{1}{a}\cdot \ln{x}}{(\ln{a})^2}$$ which is…
user359689
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What's the proof of the following formula: If f(x/y)= Constt., then dy/dx= y/x?

This formula (trick) is directly given in my study material. I have tried to prove it but its getting too long.Please help by giving proof of this condition, ie,formula.Thanks in advance.
S.Sidharth
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Simplifying $\frac{dy'}{dy}$ where $y=f(x)$

I'm feeling a bit brain-dead, perhaps it has just been a long day. But here is the problem, I think it's a simple one: I have a function $y=f(x)$, of which I then take the derivative $\frac{dy}{dx}$. Finally I take the derivative of the function…
ibell
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How do I find the derivative of $(1 +1/x)^x $

I tried one approach but the correction in the book shows me a total different answer. Here's what I did: $(1+ 1/x)^x=xln(1+1/x)$ Thus, now we try to find the derivative of a multiplication: $ u(x)=x$ $(u(x))'=1$ $v(x)=ln(1+1/x)$ $(v(x))'=…
John Mayne
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What is the difference between derevative w.r.t a vector and directional derivative?

Say we have a scalar-valued function $f: \mathbb R^3 \rightarrow \mathbb R$, such that: $$f(\mathbf x) = \mathbf x^T\mathbf a$$ $\mathbf x$ and $\mathbf a$ are two vectors. The derivative of $f$ with respect to $\mathbf x$ is $\nabla f$, but the…
Syntax_ErrorX00
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