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.

33197 questions
3
votes
2 answers

Product Chain Power Rule： Either it's the book or I am wrong.

The problem:　$x^3\sqrt{2x+4}$ $f(x):= x^3$, $g(x):= \sqrt{2x+4}$ $(f\times g)' = f^{\prime}g+fg^{\prime}$ thus it should be $3x^2\sqrt{2x+4} + (x^3)[\frac{1}{2}(2x+4)^{\frac{-1}{2}}(2)]$ which is $3x^2\sqrt{2x+4}+\frac{x^3}{\sqrt{2x+4}}$…
yiyi
  • 7,352
3
votes
0 answers

Differentiable function only at $x=n$ where$ $n is an integer

Suppose $f:\mathbb R \to \mathbb R$ is only differentiable at integer points. Is this possible? If does, what kind of function is $f$?
user197137
  • 147
3
votes
3 answers

Why's the derivative of $f(x) = x^3-5x-2 $ not $3x^2-7$?

I wanted to resolve this problem : $$ f(x) = 3 x^2 - 5 x - 2 $$ to a derivative, and I did it like this : $ \begin{align} f(x) &= x^3-5x-2 \\ f'(x) &= 3x^2-5-2 \\ &= 3x^2-7 \end{align} $ but once I checked the correction, I found…
R00t
  • 35
3
votes
2 answers

how to calculate $\frac{d\dot{x}}{dx}$

Let $x$ depend on $t$. $\dot{x}$ is derivative $x$ over $t$. I want to know if there are formulas to simplify $\frac{d\dot{x}}{dx}$? Any hint or thought is appreciated. Thank you!
ashim
  • 904
3
votes
1 answer

Derivative of ellipse

Find $y'=\frac{dy}{dx}$ when: $$2x^2-xy+y^2=1$$ How do I solve this? Do I start with this?: $$y=\frac{2x^2+y^2-1}{x}$$ $$y'=\left(\frac{2x^2+y^2-1}{x}\right)'$$
peppa
  • 217
2
votes
3 answers

derivative problem. is it same?

First derivative of $y=\ln(x)^{\cos x}$ is $-\sin x\ln x+\frac{\cos x}{x}$ or another answer? My friend gets another answer, but it's true? thanks.
user3658777
  • 61
  • 1
  • 6
2
votes
2 answers

Differentiability - general function?

I know that a function is differentiable if the limit exists as $\Delta x \to 0$ of a certain limit. But how can one know this beforehand? I mean, we usually just differentiate using rules that we have derived using the limit definition, but then…
Jack Dan
  • 21
2
votes
2 answers

differentiability of $\tan^{-1}(\frac{1}{|x|})$

How to justify, the following function is differentiable at origin or not? $f(x) = \tan^{-1}\frac{1}{|x|}$ if $x \ne 0$, $f(x) = \frac{\pi}{2}$ if $x = 0$. Even though mod x is not behaves well at the origin, since we are composing the mod x with…
GA316
  • 4,324
  • 27
  • 50
2
votes
2 answers

$\displaystyle \frac{d}{dx}2^x$ where $x=0$

I put into Wolfram Alpha: d/dx 2^x Where it told me $f'(x)=2^x\log(2)$. Then I put in d/dx 2^x where x=0 and it said "$\displaystyle \log(2)\approx0.693147$" I know through Wolfram Alpha and a couple of calculators $\log(2)\approx0.30103$. But…
Gᴇᴏᴍᴇᴛᴇʀ
  • 915
  • 2
  • 14
  • 28
2
votes
1 answer

Issue differentiating the Lambert W function

I want to differentiate the Lambert W function (the inverse of $y = xe^x$), I didn't think it would be that difficult a problem but it's causing me some problems. I tried this method: (1.) Implicitly differentiating $f(g(x)) = x$ and solving for…
user3002473
  • 8,943
2
votes
4 answers

Prove that $f'(x_o) =0$

Let $f$ be a function defined on an interval $I$ differentiable at a point $x_o$ in the interior of $I$. Prove that if $\exists a>0$ $ \ [x_o -a, x_o+a] \subset I$ and $ \ \forall x \in [x_o -a, x_o+a] \ \ f(x) \leq f(x_o)$, then…
S.Dan
  • 1,115
2
votes
6 answers

Derivatives of equations

Assume that $x$ and $y$ are related by the equation $y\ln x=e^{1−x}+y^3$. Compute $dy/dx$ evaluated at $x=1$. I do not understand how to compute the derivative of an equation. Please explain.
Minu
  • 971
  • 1
  • 9
  • 16
2
votes
3 answers

Proof of $n^{th}$ derivative Test

Proof needn't be a rigourous , but should give an insight of how $n^{th}$ derivative test (higher order derivative test) works as i know how to use it in application but i don't much understand it ,especially the inflexion point thing.
Maggi Iggam
  • 347
2
votes
2 answers

Find equation of a curve by tangent

This is probaly easy. Find the equation of the tangent to a curve $y = u (x)$ is $y =(-2a+4)(x-a)+k$, where $k$ is a constant. Given that the curve touches the $x$-axis at $x =2$, find the value of $k$ and the equation of the curve.
esoteric elf
  • 171
2
votes
0 answers

Differentiate the expression

i HAVE this expression that includes gamma functions. Differentiate the following w.r.t. 't' then taking t=0 . Any hints are welcome, because I am clueless, at present how to solve it.
SA-255525
  • 187
1 2 3
…
99 100