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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What should be the conditions for $\frac{d^2x}{dx^2} = 0$?

Let us have: $$\cases{ F = F(x,y)=const \\ y = y(x)} $$ I am taking a second differential over function $F$, i.e. $d^2F$. During calculations I have this term: $$\frac{d^2x}{dx^2}$$ I feel awkward, as I want it to be zero, but the only way I can see…
positive
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Derivative of $\arctan \sqrt x$

The derivative of $\arctan \sqrt x$ is \begin{equation} \frac{1}{2\sqrt x(1+x)} \end{equation} or \begin{equation} \frac{1}{2\sqrt x(1+|x|)} \end{equation}?
Mark
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Derivative of 'oneplus' activation function, in terms of function's output?

In machine learning there is a function called 'oneplus': $$y = 1+ ln(1+x)$$ Edit: this is actually a wrong formula. For the correct one, see the comment under this question, and the accepted answer. The derivative in terms of input is $$…
Kari
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Weird factoring out of a derivative

Is it possible to just factor out the derivative like this: $$ x \left[ \frac{d^2 \Psi}{d x^2} \cdot \Psi^* - \frac{d^2 \Psi^*}{d x^2} \cdot \Psi \right] = x \, \frac{d}{dx} \left( \frac{d \Psi}{d x} \cdot \Psi^* - \frac{d \Psi^*}{d x} \cdot \Psi…
71GA
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How to find $\frac{\mathrm{d} r}{\mathrm{d}x}$?

There are $2$ formulas, $x=r \cos \theta$ and $r=(x^2+y^2)^{1 \over 2}$ and I need to find $\frac{\mathrm{d}r}{\mathrm{d}x}$, the solution is $\frac{\mathrm{d}r}{\mathrm{d}x}=\cos \theta$. This answer told me that it considers $y$ as a constant so…
Student
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how do you differentiate x^(3/4) using first principle

$$\lim_{h\to 0}\frac{\Bigl((x+h)^{\frac{3}{4}}-(x)^{\frac{3}{4}}\Bigr)}{h}$$ I understand the process till $$\lim_{h\to 0}\frac{\Bigl((x+h)^{\frac34}-(x)^{\frac{3}{4}}\Bigr)}{h} *…
Ashwin Sarith
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Race between two dots

A moving dot P departs from O at an initial speed of 6 m/s and accelerates at 2m/s^2 in the east direction. 2 seconds after P's departure, dot Q departs from O to chase down P at a constant speed of k m/s in the east direction. What is the minimum…
user66246
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The electric current discharging in a R-C circuit is given by $i=i_0e^{\frac{-t}{RC}}$, where are $i_0, R, C$ are constant parameters.

t is time. Let $i_0=2.00 units$, $R=(6.00)10^5 unit$ and $C=(0.500)10^{-6}$. Find the rate of change of current at t=0.3s The answer need to be obtained by differentiating wrt t $$\frac{di}{dt}=[i_0][e^{\frac{-t}{RC}}][\frac{-1}{RC}$$ Inputting the…
Aditya
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Derivative for summation of two-function product

Suppose that I have $f(x) = \text{ln}\Big(1 + \sum_{i=1}^{I}p_i(x)q_i(x)\Big)$. I want to find $f'(x)$. Based on my understanding, I would obtain: $f'(x) = \frac{1}{1 + \sum_{i=1}^{I}p_i(x)q_i(x)}\frac{d}{dx}\Big(1 +…
bnbfreak
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What's the meaning of a derivative in this example?

This is the problem I was trying to solve: "Let $f(t)$ be the number of words a certain person learns per day when she is $t$ years old. What is the meaning of $f'(20)$? What are the units of measurement?" I was thinking, since $f'(t)$ represents…
user36586
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Derivative of a summation w.r.t. inside term

My question is close to this one except for one term. More specifically: $$\frac{1}{m}\sum_{i = 1}^m (x_i - \mu)^2$$ If I wanted to find the derivative of this entire term w.r.t. $x_i - \mu$, how would I go about that? Is this even…
Sean
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Is my derivation correct?

Have I done the right derivative? $\frac{d}{dx}log(2\pi x^2)^{-n/2}=\frac{1}{(2\pi x)^{n/2}} (\frac{-n}{2} (2\pi x)^{3n/2} 2\pi)$
user333750
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Clarification about this step taken in this proof of differentiation

I was looking at proofs of the differentiation of exponentiation and I'm confused about the last step of this part of the proof: $\lim_{x \to x_0} \frac{f(x)-f(x_0)}{x-x_0}$ = $\lim_{x \to x_0} \frac{x^n-x_0^n}{x-x_0}$ = $\lim_{x \to…
NotSureHowToDoStuff
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Simplification of $\frac{d}{dx}\frac{dx}{dt}$?

How can $\frac{d}{dx} \frac{dx}{dt}$ be simplified, where $x$ is a function of $t$? My guess is that since partial derivatives commute, that $\frac{d}{dx} \frac{dx}{dt}=\frac{d}{dt}\frac{dx}{dx}=\frac{d}{dt}(1)=0$, but am not sure if I'm missing…
sbr
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Differentiation of $\sin^{-1}(1/\sqrt{1+x})$

$$\arcsin \frac{1}{\sqrt{1+x}}.$$ I tried differentiating it but every time my answer came wrong. The answer in my book is $$- \frac{1}{2(x+1)\sqrt x}.$$
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