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

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1answer
20 views

Optimization of parallelepiped.

Let $K \in R^3$ the ellipsoid given by the equation $ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $ with $a,b,c > 0$ , let $(x,y,z) \in K$ on the first octant, consider the ...
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6answers
132 views

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.
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1answer
33 views

$f$ a differentiable fucntion in $[a,b]$ with $f´(a) < C < f´(b)$

Let $f$ a differentiable fucntion in $[a,b]$, suppose the existence of a point $C$ with $f´(a) < C < f´(b)$ how can i deduce that given the function $g(x) = f(x) - C(x-a)$ then exist a pint ...
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1answer
56 views

Calculating $\frac{d}{d(x^2)}f(x)$

There's a question I need to solve, which requires that I take the derivative of some function by the square of a variable, and I'm not sure how to do such a thing. For example: $\frac{dx}{d(x^2)}$ - ...
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2answers
44 views

Prove an inequality (Using Taylor expansion)

Prove: $\frac{x}{1+x} < \ln(1+x) < x$. I thought a good practice would be to prove it using Taylor Expansion. Here's my try: $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}...$$ The n=1 ...
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1answer
30 views

How can I calculate the derivative of a Catmull-Rom spline with nonuniform parameterization?

Allow me to preface this by saying I am not a trained mathematician in any sense, so it's entirely possible I'm missing something rather fundamental. That said, I'm trying to take the derivative of a ...
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0answers
18 views

Prove or disprove the statement related to the definition of multivariable-differentiable function

The question: Let $f,f_1,...,f_n \; (n > 0)$ be functions from $\mathrm{D} \subset\mathbb{R}^n$ to $\mathbb{R}$ satisfying $$\left ( \sqrt{\sum_{i=1}^n x_i^2} \right ) f(\mathrm{x}) = \sum_{i=1}^n ...
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3answers
40 views

Derivative of sum of two exponential functions

I have the following formula - $$ f(x) = \left(0.1 e^{-1.5{x}^{0.2}} + 0.9 e^{-0.5{x}^{0.1}}\right)^{c}$$ where $\bf c$ is a constant value. How can I solve $f'(x)$ ? According to the answer, I ...
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1answer
19 views

Maximize area of a rectangle between parabola and a line

I was given a task to maximize the area of a rectangle that can be inscribed between parabola $y=1-x^2$ and a line $y=0$ such that one side of the rectangle lies on the $x$ axis. My idea is to somehow ...
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1answer
37 views

Why is $(\sec x)' = \tan x\sec x$ and not $\tan x$?

As far as I understood, the Fundamental Theorem of Calculus states that the integral of a function is its anti-derivative. And yet, although the integral of $\tan x$ is $\sec x$, the derivative of ...
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4answers
40 views

derivative $\frac{df(t)}{dt}$ of $f(t) = \int_0^t\ln{(s^2+t^2)} ds$

Let $f(t) = \int_0^t \ln{(s^2+t^2)} ds$, how can I find the derivative $\frac{df(t)}{dt}$? The function $\,\int_0^t \ln{(s^2+t^2)} ds$ is defined to be continuous in $s^2+t^2 > 0$ and $ s^2+t^2 ...
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4answers
79 views

Local minimum of $f(x) = 4x + \frac{9\pi^2}{x} + \sin x$

What's the minimum value of the function $$f(x) = 4x + \frac{9\pi^2}{x} + \sin x$$ for $0 < x < +\infty$? The answer should be $12\pi - 1$, but I get stuck with the expression involving both ...
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0answers
29 views

Last step in derivation of Euler-Lagrange equation (definite integral)

In the classical derivation of Euler-Lagrange equation in the calculus of variations, for a case with fixed end points at $x=a$ and $x=b$, we have the final step in derivation arriving at: ...
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1answer
108 views

showing $\int _a^b\left(f'\left(x\right)\right)dx\:=\:f\left(b\right)-f\left(a\right)$

Let $f(x):[a,b]\to \mathbb R$, be differentiable on $[a,b]$ (and continuous) so that $f'(x)$ is integrable on $[a,b]$. I need to show that: $$\int _a^b\left(f'\left(x\right)\right)\mathrm dx = ...
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1answer
20 views

Direction for greatest derivative

Suppose I have a function like $f(x,y) = e^x e^y x^2 y^2$, and I want to know in which direction the derivative will grow fastest at a stationary point. $(0,0)$ is a stationary point of the example ...
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1answer
30 views

Finding tangent line of $f(x) = 1/x$

Find the equation of the tangent line to $f(x) = \dfrac{1}{x}$ through the point $(0, \alpha)$. Answer: $y = −\alpha^2\dfrac{x}{4} + \alpha$ I've found $f'(x)=-\dfrac{1}{x^2}$ but how do I find ...
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2answers
46 views

Derivative function went wrong

I am trying to take the derivative of this function but I am facing some difficulties. $$f(x)= e^{\ln(e^{7x^2+11})}$$ My answer was : $7e^{(7(x^2))}*14x$ I cancelled the $\ln$ with the $e$ first, ...
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2answers
68 views

Prove $\sqrt{x}>\ln(x)$ in $[1,\infty)$

Well, i try to prove this statement. i choose to make function: $f\left(x\right)\:=\:\sqrt{x}-\ln x$ but the derivative is: $\dfrac{\sqrt{x}\:-\:2}{2\sqrt{x}}$ and it's not always greater than $ 0$. ...
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0answers
74 views

general solution of the equation $\frac{dy}{dx} =\exp(y/x)$

How can i get the general solution of the equation a) $\frac{dy}{dx} = \exp(y/x)$ b) $\frac{dy}{dx} = \exp(x-y)$ and $y=2$ when $x = 0$ I tried b) first: This is a first-order nonlinear ordinary ...
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2answers
33 views

Derivative of rational function help.

consider $$f(x)=\frac{1}{2x-4}$$ The derivative should be $\displaystyle -\frac{1}{2(2x-4)^2}$ However I get $\displaystyle -\frac{2}{(2x-4)^2}$ my workflow: $$\begin{array}{} f'(x)&= ...
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0answers
29 views

Deriving stationary points using the second order derivative.

Suppose that for some function $f$ we want to know the stationary points, i.e. $\frac{\partial f(\mathbf{x})}{\partial \mathbf{x}} = \mathbf{0}$. We can define a new function ...
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3answers
96 views

differentiability check

$$f(x)=\frac{1}{x-2}$$ number of points where $f$ is not differentiable? I know that the domain of the function is $\mathbb{R}\setminus\{2\}$ and differentiability is checked only in the domain of ...
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0answers
21 views

What is the 1st derivative i.r.t. coordinates for a vector function?

For a vector function $f(x,y,z)$, we have the divergence $$\nabla \cdot f(x,y,z) = \frac{\partial{f}_{x}}{\partial x}+\frac{\partial{f}_{y}}{\partial y}+\frac{\partial{f}_{z}}{\partial z}$$ , the ...
1
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1answer
34 views

$F(x) = \int_0^x \sin{((x+t)^s)} dt$

Let $F(x) = \int_0^x \sin{((x+t)^s)} dt$ , how can i find the derivative with respect to $x$. First i tried to use the fundamental theorem of calculus that asserts that $$\text{if } F(x) = \int_a^x ...
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2answers
28 views

Slope of the tangent line in Calculus

Find the slope of the tangent line to $c(x) = e^{g(x)}$ $$g(x) = 2\:,\:g'(x) =-2$$ I tried and my answer was $-14.7781122$ but it was wrong ! why ?
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4answers
54 views

Calculus about derivative

How to solve this derivative? $$\large p(t) = 3e^{{-2e}^{2t}}$$ It looks weird to have two exponents instead of one. I tried to solve it but i got stuck.
3
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1answer
44 views

Finding limit of cube root [duplicate]

I'm trying to evaluate this limit, but I don't think it's coming out correctly. Could someone please offer me some assistance? Evaluate limit analytically $$\lim_{h\to 0}\frac{\sqrt[3]{x + h} - ...
1
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1answer
37 views

Find the partial derivatives of second order of $f(x,y)=\varphi(xy,\frac{x}{y})$

Ok guys, I'm given this smooth function $\varphi(u,v)$ defined in $R^2$. So that $f(x,y)=\varphi(xy,\frac{x}{y})$. I have to find all partial derivatives of second order of $f$ using the partial ...
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1answer
19 views

Solving an ODE using variations of parameters and Wronskian theorem.

So I am attempting to solve this differential equation by trying to follow an example that my professor did in class. I am just not too sure about my answer seeing as WolframAlpha gives me this: ...
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2answers
65 views

Local minimum implies local convexity?

Consider a real function $f$, and suppose it has a local minimum at $a\in \mathbb R$. It typically looks like What hypotheses can be added to $f$ so that there is some $\epsilon >0$ such ...
3
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0answers
53 views

Roots of derivative of q-expontial function

Let the q-deformation of the exponential function be defined by $$ e_q(z)=\sum_{n=0}^\infty{\frac{z^n}{[n]_q!}}. $$ Eq. (1.8) of this paper provides the product representation $$ ...
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1answer
22 views

finding velocity from a table

I have a homework question I am seeking an alternative solution to. Basically, the question is... "The table provided below shows the position of a particle S, at several times, t. as the particle ...
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1answer
33 views

if $f$ is differentiable at $x_0$ then the limit exists

Let $f$ differentiable at $x_0$. Show that the following limit exists $$ \lim_{h\rightarrow0} \frac{f(x_0+h)-f(x_0-h)}{h}$$ If $f$ is differetiable at $x_0$ then it's one-sided derivative exists ...
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2answers
70 views

Derivative of $f(x)=|x|$

Okay, so $\displaystyle \frac{d}{dx} |x| = \frac{|x|}{x}$. But I have trouble seeing why. Here's what I've tried: $$\frac{d}{dx}|x|=\begin{cases} \frac{d}{dx}x & \text{if }x > 0 \\ ...
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1answer
54 views

$F(x) = \int_0^{x} t^2 e^{t^2}dt$

Let $y_0 = f''(2) + f'(1) + f(0)$ if $f$ is a real function defined by $f(x) = \int_0^{x} t^2 e^{t^2}dt$. How can I calculate the value of the expression $y_0$. I tried use the fundamental theorem ...
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4answers
47 views

$f(x,y) = g(\sqrt{x^{2}+y^{2}})$ Prove that f is differentiable at $(0,0)$ iff $g'(0)=0$

$f(x,y) = g(\sqrt{x^{2}+y^{2}})$. Prove that f is differentiable at $(0,0)$ iff $g'(0)=0$ This was a question on my midterm a few days ago. I've been thinking about it for a while and still cannot ...
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0answers
9 views

Implementation of Total Variation Regularization Algorithm (Lagged Diffusivity Algorithm)

I am trying to compute the derivative of an experimentally-measured quantity as a function of time. The data are fairly noisy, which causes problems. For instance, using finite differences (central ...
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1answer
81 views

Prove there's $x_0$ such that $f'(x_0)=0$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ differentiable at $\mathbb{R}$ and: $$\lim_{x\rightarrow \infty}\left( f(x)-f(-x) \right) = 0$$ Show there's $x_0$ such that $f'(x_0) = 0$. I tried to use ...
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1answer
21 views

Derivative of diagonal function

I'm working on a sightly modified least-squares method which must minimize the quantity: $$ [Y-\text{diag}(\mu X^T)]^T\cdot [Y-\text{diag}(\mu X^T)] $$ where $Y$ is a $n$-dimensional vector and $\mu$ ...
3
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1answer
51 views

Prove that a function is differentiable if…

I'm trying to prove that given a differentiable function $f: \mathbb{R}^2 \to \mathbb{R}^m$ in $p =(p_1, p_2) \in \mathbb{R}^2$, the function $$ g(x, y) = f(x, y) - \frac{\partial f}{\partial x}(p)(x ...
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1answer
73 views

Local Maxima of Piecewise Function [closed]

$$f(x)=\left\{\begin{array}{l l}x+1 & \text{for }x<0,\\ 1 & \text{for } x \ge 0 \end{array}\right.$$ Will the local maxima of the function $f$ be at $x=0$? Will the global maxima of the ...
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4answers
46 views

Differentiate $\frac{\ln(x)}{\cos(x)}$

Please help me with this question. $$y= \frac{\ln(x)}{\cos(x)}$$ Just starting with calculus. Thank you.
1
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1answer
40 views

Rates of change question?

A boat is observed from top of a $100\ \text m$ high cliff. The boat is travelling towards the cliff at a speed of $50\ \text{m/min}$. How fast is angle of depression changing when angle of ...
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2answers
24 views

derivative with respect to a vector/matrix

please excuse the stupid question but I cant find anything online.. If $$f(\vec{x}) = \vec{x}^TA\vec{x}$$ with $A$ being a matrix, then $$ \frac{df}{d\vec{x}} = \vec{x}^T(A+A^T)$$ Can someone tell ...
2
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3answers
46 views

$ F(x) = \int_0^2 \sin(x+l)^2\ dl$

Consider the function : $ F(x) = \int_0^2 \sin(x+l)^2\ dl$, calculate $ \frac{dF(x)}{dx}|_{x=0}$ the derivative of $F(x)$ with respect to $x$ in zero. Let $g(x) = \sin (x)$ and $h(x) = (x+l)^2$ then ...
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1answer
15 views

Derivative of exponential function

1) $f(t) = (\ln 5)^t$ what is the $f'(t)$? I tried $t\ln(5)$ but it was wrong. 2) $f(x) = x^{\Large π^6} + (π^4)^x$ This one I did not attempt in it because I find it confusing little bit.
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3answers
183 views

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 ...
6
votes
2answers
161 views

simple way to show $|| \partial_x \int_{B(x,\epsilon)} \frac{x-y}{|x-y|^3} f(y) dy||_{\infty} = O(||f||_{\infty})$ in $\mathbb{R}^3$

We are set in $\mathbb{R}^3$. Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ be a $C^1_0$ function, i.e. continuously differentiable with compact support. Let $\epsilon > 0$ be small. I need to show ...
0
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1answer
54 views

$y(x) = \int_0^x \frac{\sin(t)}{t}dt $

Let $y(x) = \int_0^x \frac{\sin(t)}{t}dt $ find maximums and minimums of $y(x)$. First let $F(x) = \int_0^x \frac{\sin(t)}{t}dt$ and $f(t) = \frac{\sin(t)}{t}$ then $F'(c) = f(c) $ then if $ ...
0
votes
1answer
100 views

$f(x,y) = (1- \cos(\frac{x^2}{y})) \sqrt{x^2+y^2}$

Let $f(x,y) = (1- \cos(\frac{x^2}{y})) \sqrt{x^2+y^2}$ for $y \ne 0$ How can I prove that f is not differentiable in $(0,0)$. Please some help.