Mathematics Stack Exchange
Mathematics Stack Exchange

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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How to solve $n$-th derivative of logarithm function?

How to slove the $n$-th derivative of logarithm function $$\frac{d^n}{dx^n} \ln \left(\frac{p}{ax+b}+\frac{1-p}{b}\right) $$ where $p \in [0,1]$, $a$ and $b$ are positive numbers
Hui Zhao
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what is the nth-derivative in 0 of $\frac{e^x}{1-x}$

Using $$f^n(0)=n! .a_n$$ I have$$\frac{e^x}{1-x}=\sum{\left(\frac{x^n}{n!}\right)}\sum{x^n}=\sum_{n \geq 0}\left({\sum_{k=0}^{n}{\frac{n!}{k!n!}}}\right)$$ How to get the combination?
mezzaluna
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Show that $(\frac{\partial u}{\partial s})^2 \neq \frac{\partial^2 u}{\partial d s^2}$

Assuming that $u = f(x,y)$, $x = e^s\sin(t)$, $y = e^s\sin(t)$ Show that $(\frac{\partial u}{\partial s})^2 \neq \frac{\partial^2 u}{\partial d s^2}$ I know what to do, but I don't know how to do it. The RHS gives me difficulties.
Arnold
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A Method for Finding the Expansion of $\sin m\theta$ and $\cos m\theta$

A method to find the expansion of a function is to first differentiate the equation $y=f(x)$ twice with respect to $x$, and combine the results to form an equation in $y$, $y'$, and $y''$. Next, assume $y=A+Bx+Cx^2+\&\text{c}.$ Differentiate twice,…
Crescendo
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Prove that $f$ is differentiable if and only if $f^i$ is differentiable for $i=1,...,m$

Let $f= \begin{pmatrix} {f^1} \\{...} \\{f^m} \end{pmatrix} $ where $f^i : D \to \mathbb R$. Prove that $f$ is differentiable if and only if $f^i$ is differentiable for $i=1,...,m$. I feel like this is straightforward, but I'm a little stumped…
user415105
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Derivative of function = indeterminate form

How can I calculate the derivative in $x=0$ of $$f(x)=(x-\sin x)^{1/3}?$$ I found the derivative and replaced $x$ but it would be an indeterminate form and if I try using limit of $(f(x)-f(0))/x$, it doesn't lead me anywhere so how could I find it?
Lola
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what's the derivative of $\cos(x) \sin(x)$

before I ask for anything I must admit I'm working hard to understand this beautiful subject. Thanks in advance. I want to get a derivative of: $\cos(x)\sin(x)$ The solution is the following: $$ \frac d{dx}(\cos(x) \sin(x)) = \cos(2 x)$$ Where…
Diego Pacheco
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Question about Quotient rule and chain rule

I have the problem: Find the Derivative: $$\frac{4y^6-6y}{e^{4y}+y}$$ I used the quotient rule $$ \left( \frac{f}{g}\right)' = \frac{f'g-fg'}{g^2}$$ After deriving, I got $$\frac{(24y^5-6)(e^{4y}+y)-(4y^6-6y)(4e^{4y}+1)}{(e^{4y}+y)^2}$$ Do I need to…
ASTR
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Finding a differentiable function with certain properties

I have to find a differentiable function $f: \mathbb{R} \to \mathbb{R}$ with $f'(x)=0$ if $x < 0$ and $f'(x)=1$ if $x≥0$. I think that such a function doesn't exist because the left and right limit for $x \to 0$ are different. Can I proof it like…
Mee98
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First variation in three dimensions

Consider the following problem from fluid mechanics: $\phi(\rho,q)$ is a function dependent on density and potential vorticity $q:=\Omega\cdot \nabla \rho$ with $\Omega=curl (v)+ f$ and $f$ being independent of $v$ and $\rho$ Now I would like to…
Master
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Show that $f$ is differentiable and find the formula of $Df(I)(H)$.

Consider the function $f:M_{n\times n}(\mathbb R)\to M_{n\times n}(\mathbb R)$ with the formula given below: $f(A)=A^t+A^2A^t$ Show that $f$ is differentiable and find the formula of $Df(I)(H)$. ($I$ is the identity function.) So, i was…
Paolo Molina
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What's wrong with the following differentiation?

Consider $f(x)=x^2$ Then $f'(x) = 2x$ But $f(x) = x+x+x + ...+x$ ($x$ times) then $f'(x)=1+1+1+...+1$   i.e. $f'(x)$ $=x \ne 2x$ So this implies that if we see derivative as an operator, its proper usage needs that the term that…
maverick
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Finding the nth derivative of a function

Recently I am working on a project. It requires the nth derivative of the function $e^{i\frac{1}{\sqrt{1+\lambda}}z}$ with respect to $\lambda$. Or equivalently, finding the nth order Taylor series near $\lambda=0$ to nth order. Using mathematica to…
Siyi Zhou
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$f'''(0)$ of $f(x)=\sin x/x,x \neq 0,1, x=0$

I want to find the Taylor polynomial at zero for $f(x)=\sin x/x \ \text{if} \ x \neq 0, 1 \ \text{if} \ x=0$ and struggle to find $f''(0).$ I can't just differentiate $\sin x/x$ because we are considering $0$.I have $f'(0)=\lim_{x \to 0}…
user30523
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$f(x){d^ng(x)\over dx^n}= \sum_{k=0}^{n}(-1)^{k}(_{k}^{n}){d^{n-k}\over dx^{n-k}}(f^{k}(x)g(x))$

Rule $f(x){d^ng(x)\over dx^n}= \sum_{k=0}^{n}(-1)^{k}(_{k}^{n}){d^{n-k}\over dx^{n-k}}(f^{k}(x)g(x))$ Above rule looks transformation of Leibniz's rule. Tried to prove it but has a problem to transform the parameters of Combination. Anyone can prove…
Beverlie
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