Questions tagged [derivatives]

Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

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12 views

How to calculate the partial derivative function at any general point?

screenshot with transcription below: Consider the function $f: \mathbb{R}^{2} \longrightarrow \mathbb{R}$ given by $$ f(x, y)=\left(1-\cos \frac{x^{2}}{y}\right) \sqrt{x^{2}+y^{2}} $$ for $y \neq 0$ ...
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23 views

Determining the times at which the extrema of trigonometric functions (eg, $170\sin120\pi t$) occur

I had a question that I just could not seem to figure out. I have been working on trigonometric functions and I was asked to find the maximum and minimum of a certain function and the times at which ...
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Derivative question requring one to draw a graph!

Is it possible to draw a function such that $f$ satifies: $f(0)=1$ $f'(x)<0$ for all real $x \neq0$ $f''(0)=0$ $f'(0)=$ undefined $f''(x)>0$ for all real $x > 0$ $f''(x)<0$ for all real $...
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Question about a simple derivative

Let's say we have $y = 10^2$ How much does $y$ increase when $x$ increases by $1$ unit? We have the form $y=x^2$ and $\dfrac{dy}{dx}=2x \ldots$ here $x=10$, so why isn't the answer $2(10)=20$? By ...
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23 views

Chain Rule in Leibniz and Lagrange notation: questions

I'm learning multiple applications of the chain rule and the notation surrounding it. Does the following notation with example correctly reflect the chain rule in both Lagrange and Leibniz notation? $...
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1answer
20 views

Exponential With Step Function

I would like to know how to differentiate the following function: $f(t)=e^{{-5}(t-2)}$. I think this function exists at $t=0$, but the derivative is valid only for $t>0$ or $t=0^+$. $$f'(t)=-5e^{{-...
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34 views

Understanding a specific process of finding the derivative of $x^TAx$

I am referring to @copper.hat's response to : Derivative of Quadratic Form. I do not have the reputation to reply directly. My goal is to find a way to better differentiate and understand these ...
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The angle of the intersection of two graphs

I encounter some problems with understanding the explanation of the follwing problem: For which value of $\alpha$ does the graph of $f_\alpha(x)=e^{\alpha x}$ intersect the graph of $g(x) = \sqrt{x+1}...
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52 views

Possible Maximum or Minimum??

I was solving some curve sketching questions and wanted to ask something that I am a little confused about. Why, when we find the first derivative of an equation and set it equal to zero to find the ...
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How to obtain the partial derivative of a Gaussian mixture with respect to a covariance matrix?

This is my attempt to obtain the partial derivative w.r.t the covariance, following hints in section 9.2 of "PRML, Bishop". Though the partial derivative is not provided in the book, my ...
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Find the derivative of the function v that is expressed in terms of u

Consider a financial economy with one physical good, two equally probable states of nature at date $1$, and $I$ consumers with linear-quadratic preferences given by the utility function $u_i: \mathbb{...
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Differentiation with respect to non-independent variable

If a function is given as $F(x(t), x'(t), t) = \int(E+V)dt$ would $dF/dE$ equal zero? We know that E and V is made up of $x, x', t$ but F is a function of $x, x', t$ not $E$. So the confusion is that ...
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31 views

Derivatives of general absolute value functions

Let $f : (a, b) \to \mathbb{R}$ be differentiable at $c \in (a, b)$, and $f(c) \neq 0$. How do I show that $|f|'(c) = f'(c)$ if $f(c)>0$? I know that $|f|'(c) = $ $\frac{|f(x)| - |f(c)|}{x-c}$. ...
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What is the derivative of f(f(x)) at a certain point? [closed]

I encountered some problems with understanding why do we need to derivate the tangent in the following exercise. Can someone please explain me? This is a screenshot of the problem and its explanation:
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Confusion in study of derivative of a piecewise function

I have some confusion about the study of the derivatives in $x=0$ of this piecewise function $$f(x)=\begin{cases} \exp\left(-\frac{1}{x^2}\right), \ \text{if} \ x>0 \\ 0, \ \text{if} \ x \leq 0\end{...
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Possible point of inflection?

Heyy guys! I have been solving multiple kinds of curve sketching questions when it comes using the first derivative and second derivative. I was curious to know why we call the point of inflection a ...
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40 views

find the n-th derivative of e^(ax) cos (bx) [closed]

N-th Derivative I used leibnitz theorem to solve it half taking u=e^(ax) and v = cos bx but stuck at the final result Finding the nth derivative of it.
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66 views

Can $ \int \sin(x)+\sum^{\infty}_{n=1} (-1)^n\frac{\sin^{(2n+1)}(x)}{(2n+1)!} dx$ be evaluated in terms of elementary functions?

How do you go about integrating this and can it even be done with elementary functions? $$ \int \sin(x)+\sum^{\infty}_{n=1} (-1)^n\frac{\sin^{(2n+1)}(x)}{(2n+1)!} dx. $$ I understand the concept of $$ ...
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35 views

How to find critical points of definite integral

Say I have a function $$g(x) = \int_a^b (f(t)-x)^3dt$$ how would I go about finding the critical points of this function? I know that FTC gives you that if $$h(x) = \int_0^x f(t)dt$$ then $$h'(x) = f(...
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Checking differentiability at $(0,0)$

Check $$f(x,y) = \begin{cases} \frac{x^3y^3}{x^2+y^2} & x^2+y^2 \neq 0 \\ 0 & x = y = 0 \end{cases}$$ is differentiable at $(0,0)$ or not . I have a very vague understanding of ...
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What type of kernel has eigenfunctions with orthogonal derivatives (Mercer's theorem)

Suppose $K(x,y)$ is a continuous non-negative definite function, where $x,y\in[0,1]$. By Mercer's theorem, $$K(s,t)=\sum _{j=1}^{\infty }\lambda _{j}\,e_{j}(s)\,e_{j}(t)\,,$$ where the $e_j$'s form an ...
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61 views

Why is the function differentiable in the point $(0,0)$?

I am trying to figure out why my function is differentiable and therefore continuous in the point $(0,0)$ which is also a critical point and a saddle point. Considering my function: $f(x, y) = x^3 - ...
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32 views

Solution of ODE $3 d^2y/dx^2+4x dy/dx-8y=0$

I have the following ODE before me: $$ 3y''+4 xy'-8y=0.$$ The question says that the integral is a polynomial in $x$. But I am finding it difficult to prove that. Please suggest.
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For which values of $m$ the line is tangent to the quadratic curve?

For which values of $m$, the line $y=2x-4$ is tangent to the curve $y=(m+3)x^2+mx ?$ We have a quadratic equation. the equation of slope of tangent line to it for specific $x$ can be find by $y'=(2m+6)...
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26 views

How would you compute that derivative involving $R\in SO(3)$?

For two vectors $u,v$ of the unit sphere $\mathcal{S}$ of $\mathbb{R}^3$, let $\DeclareMathOperator{\hat}{hat}$ $$R = (u\cdot v)I_3 + \hat(u\times v) + \frac{1}{1+u\cdot v}(u\times v)\otimes (u\times ...
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question about calculus $f(x)=e^x(x^2+x)$, derive $\frac{d^n\,f(x)}{dx^n}$

$f(x)=e^x(x^2+x)$, derive $\dfrac{d^n\,f(x)}{dx^n}$ may use Leibniz formula but i'm not sure:(
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How to find the $n^{th}$ derivative of $\dfrac{x^2-1}{(x-1)(x-2)(x-3)}$ [closed]

I am confused with the numerator part what to do after finding the partial fraction of the denominator please help. I have an exam coming up in a week
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Non local differential equation

I am trying to solve a non local differential equation of the form: $$q'(x)=kx^{k-1}q(x+c)-kx^{k-1}q(x)+\frac{k-1}{x} q(x)$$ The case $k=1$ is simple and the solution is an exponential with properly ...
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36 views

Are differentiating complex functions wrt $x$ and $y$ same?

Solution verification tag because profs make mistake and I would like you guys to respectfully take a look at this solution. If you can choose to differentiate this function of $z$ wrt $x$, then can ...
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How to prove that $2xy$ and $x^2 + y^2$ are continuous on $\mathbb{R^2} \smallsetminus \{(0, 0)\}$ by using the limit definition of a derivative?

For school (real analysis) I have an assignment where a little part is about proving that $f$ is continuous on $\mathbb{R^2} \smallsetminus \{(0, 0)\}$: $$f(x, y)=\frac{2xy}{x^2 + y^2}$$ By using the ...
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33 views

Gradient Descent on differential equation

I have a differential equation of the form $$\frac{d}{dx}f=f^2$$ I want to find a root of the second derivative of $f$, in order to maximize the derivative $df/dx$. I could of course simply solve the ...
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44 views

Quick Average Speed Question

I was studying for a test and found the following prompt (similar to another question that was asked, I just had one question that had a number of different prompts that centered around the main ...
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Quick Acceleration Vector Question

So I was studying for a test and found the following prompt: "A student programs a robot to move across a flat surface over the time interval $0≤t≤20$ seconds. The position of the robot at time $...
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Why is the integral of the derivative of f over f equal to zero over closed curves? [duplicate]

I'm trying to show that if $f(z)$ is analytic an $|f(z) - 1| < 1$, then $\int_\gamma \frac{f'(z)}{f(z)}dz = 0$ over all closed curves $\gamma$. Presumably, I need to show that this is an exact ...
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Area between curves $r=5\sin(3\theta)$ and $r=8\sin(3\theta)$

So I was studying for a test and came across the following prompt: "What is the total area between the polar curves $r=5\sin(3\theta)$ and $r=8\sin(3\theta)$?" I guess I'm a bit confused ...
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28 views

Why is the derivative of f equal to the sum of its partials along its components?

I was trying to understand the development of the solution in this answer, where $\overline{f(z)}f'(z)dz = (u - iv)(\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial y})(dx + idy)$. The ...
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1answer
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Find an equation of the line tangent to the graph of 𝐶 at the point where 𝑡=1

So I was given the following prompt when studying for a test: "A curve $C$ is defined by the parametric equations $x(t)=3+t^2$ and $y(t)=t^3+5t$. Find an equation of the line tangent to the graph ...
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1answer
25 views

Find the derivative of the fucntion using difference quotient

We define $f(x)$ = $x^{1/5}$ I applied the formula and got the following result: $f'(x)$ = $\lim_{h \to 0}$ $\frac{f(x+h) - f(x)}{h}$ After plugging in the respective values, this is what I ended up ...
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A doubt about differentiability and increasing function

Suppose $f:(a,b)\rightarrow (a,b)$ is differentiable on $(a,b)$ and for an $x_{0}$ such that $a<x_{0}<b$ , $f'(x_{0}) >0 $ then is $f$ increasing in some neighborhood of $x_{0}$? I have seen ...
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When is $g(t):=\inf_{s \in (0,t)} f(s)$ differentiable?

Under which non-trivial conditions is the function $$g(t):=\inf_{s \in (0,t)} f(s)$$ differentiable? Would the differentiability of $f\colon \mathbb{R} \to \mathbb{R}$ be enough? Where can I find some ...
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A question about the derivative of trace involving Hadamard product

Assume that $X\in\mathbb{R}^{n\times n}\geq 0$, $Y\in\mathbb{R}^{n\times k}\geq 0$ and $Z\in\mathbb{R}^{k\times n}\geq 0$. Let us define the function $f(Y,Z)$ as follows: $$f(Y,Z)=\Vert X-YZ\Vert_W^2:=...
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Give an example of a function such that the symmetric derivative $f^∗(0)$ exists but $\displaystyle\lim_{x\rightarrow 0} f(x)$ does not exist. [closed]

For a given function $f(x)$ define $f^*(x) = \displaystyle\lim_{h\to0}\frac{f(x + h) - f(x - h)}{2h},$ wherever this limit exists. $\:$ Give an example of a function such that $f^*(0)$ exists but $\...
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If 𝑓 is an odd function and lim𝑥→0 𝑓(𝑥) exists then prove that lim𝑥→0 𝑓(𝑥)=0.

a/ Give an example of a function such that 𝑓∗(0) exists but lim𝑥→0 𝑓(𝑥) does not exist. b/ If 𝑓 is an odd function (i.e., 𝑓(𝑥)=−𝑓(−𝑥) for all 𝑓∗(0)=0) and lim𝑥→0 𝑓(𝑥) exists then prove ...
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1answer
55 views

Give an example of a function such that 𝑓 '(0) exists but lim𝑥→0 𝑓(𝑥) does not exist.

Give an example of a function such that $f’(0)$ exists but $$\lim_{x \rightarrow 0}f(x)$$ does not exist. Hello, I am struggling to find an example of this? Much help, thanks!
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34 views

Derivative of trace involving Hadamard product

Let us assume that $A, S\in\mathbb{R}^{n\times n}$, $U\in\mathbb{R}^{n\times k}$, and $V\in\mathbb{R}^{n\times k}$. I am trying to differentiate the following expression: $$\Phi(U,V)=\mathrm{trace}\...
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53 views

Differentiating $\frac{1}{7}\cos^4(x^2+2x+3)$

I'm trying to differentiate this function using the chain rule, but not sure whether I'm doing it right. I did the following: I let $u = (x^2 +2x+3)$ and $y = \cos^4u$, $\frac{du}{dx} = 2x +2$, $\frac{...
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67 views

Let $a_n=f(1/n)$. Show that $a_n$ converges

Let $f:(0,1] \mapsto \mathbf{R}$ be a differentiable function with $|f'|\le 1 \ \forall x\in (0,1]$. Let $a_n=f(1/n)$ be a sequence with $n \in \mathbf{N}$. Show that $a_n$ converges. I know that ...
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39 views

Partial Derivative Percentage Problem

Electrical power $P$ is given by $P = V^2/R$ where $V$ is voltage and $R$ is resistance. Approximate the percentage error in calculating power if the percentage errors in measuring $V$ and $R$ are $2$%...
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2answers
55 views

A Basic Limit From Exponentials [duplicate]

Reading the proof of exponential derivatives I understand this: To show that $(2^x)'=\ln 2 \cdot 2^x$ in the proof is used the limit: $$\lim_{x \to 0} \frac{2^x-1}{x}$$ My question is: ¿How do I prove ...

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