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

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Compute the following norm derivatives

I was wondering if anyone can explain me how to compute the derivatives of the following norms: $\frac{d}{ds}||x+sp||^2_q$ for $x,p\in\mathbb{R^n}$ and $1<q<\infty$ $\bigtriangledown ...
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3answers
51 views

Mean Value Theorem Confusion.

In H. Cartan's Differential Calculus, Theorem 3.1.1 is called the Mean Value Theorem and is stated as: Theorem: Let $f:[a,b]\to\mathbf R^n$ and $g:[a,b]\to\mathbf R$ be two functions which are ...
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1answer
111 views

How to differentiate this negative power?

I'm reading the book "Calculus made easy" and I'm stuck with a step of a derivative with a negative power. Here is what is in the book: Case of a negative power. Let $y=x^{-2}$. Then proceed ...
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1answer
13 views

Using unbounded derivative to show function is not uniformly convergent

I'm confused how to use the following theorem: 19.6 Theorem. Let $f$ be a continuous function on an interval $I$ [$I$ may be bounded or unbounded]. Let $I^◦$ be the interval obtained by removing ...
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0answers
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A Maths o levels qn [on hold]

The height above ground level h m, of a capsule on the Singapore Flyer is modeled by the equation , h = 80(1-cos kt) , where k is constant and t is the time in minutes after starting the ride at ...
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29 views

Proof of taking derivatives of both equal sides

I am curious about the proof of the following or whether the statement is true in general Assume that I have the following property: $f(x,y)=g(x,y,z)$ Can I assert that $D_xf=D_xg$ at any point ...
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1answer
10 views

invertibility, derivative, and difference quotient

Suppose that $f$ is an invertible differentiable function, that the domain of $f^{-1}$ contains an interval around $a$, and that $f^{-1}$ is continuous at $a$ and that $f^{-1}$ is continuous at $a$. ...
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4answers
24 views

Derivative of $y=\tan(3)e^x$,

If, $y=\tan(3)e^x$, wouldn't the derivative be $y\;'=\sec^2(3)e^x \times e^x$? The outer function times the inner function, using the chain rule? The answer key gives the derivative as $y=e^x \tan ...
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1answer
81 views

L'Hopital's Rule, Factorials, and Derivatives

I have the following limit $\displaystyle\lim_{n\to\infty}\frac{e^n}{n!}$. Now if I try to solve this using this using L'hopital's rule, I won't be able to since I can't take the derivative of $n!$. ...
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1answer
10 views

Non-singular derivative definition

I have a basic definition question. I am studying inverse function theorem, and I am stuck with what it means to say that for a $f'$ is non-singular? I looked it up in the internet, but it did not ...
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2answers
31 views

Struggling to find the second derivative of this function's first derivative

So I've found the first derivative of this function but now I have to find the second derivative. I've tried everything but I cannot seem to get it. Here's the original function: $x = a sec(θ)$, $y = ...
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Differential Approach

Use differential to find the approximate value for ( 4,001 )^( 2 / 4,001 ) I don't know how to take the value of x for make the normal approach when we have just sqrt( 65 ) for an example.
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15 views

how to show the concavity of a very complex function?

To prove that S(n) have local maximum, I am thinking of taking second-order derivatives to $n$, then discussing the other parameter values. But S(n) seems way too complicated to me, I was wondering if ...
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3answers
45 views

Find derivative of $f(x)=\frac{1}{\sqrt{x+2}}+2x$ by definition

Use the definition of a derivative to find the derivative of: $$f(x)=\frac{1}{\sqrt{x+2}}+2x$$ my work: $$\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$ ...
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1answer
16 views

Derivative of complex conjugate

In general, two different mathematical operations need not commute. Let f(x,y) be a complex valued function, taking in two real-valued inputs x and y. Then under what circumstances is the partial ...
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1answer
13 views

how to find the interval at which a derivative function is increasing

Alright, so here's the deal. I need to find the interval of this derivative function: f(x)= −5x2+12x−7 So far, I've gotten that the derivative is this: ...
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0answers
15 views

Derivative of vectors dimension do not agree

I have two n by 1 vectors $\mathbf w,\mathbf v$ with respect to $\mathbf w$, and $\mathbf v$ is some function to $\mathbf w$. so I can get a scalar from $\mathbf w^T\mathbf v$, I want to take ...
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2answers
27 views

Fundamental theorem of calculus problem - trig functions

My problem is: On the interval (0 , pi/2). I know I need to split it in two integrals, but I don't know how. I would appreciate any suggestions on how to proceed.
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1answer
17 views

Rate of Change with Derivatives

We just started with rate of change while using derivatives and I am stuck on a question, hope you can help. A particle moves on a vertical line so that its altitude at time $t$ is $y=t^3−12·t+3$, ...
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0answers
24 views

Area of a circle shall equal the area of a square [on hold]

How can I, using bolzanos theorem, discuss the equal areas of a circle and a square? How can this be shown in a graph? Would be really grateful if any could help me! :)
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2answers
44 views

Why is $y{(\log_a(x))} = \log_a{(x^y)}$?

Why is $y{(\log_a(x))} = \log_a{(x^y)}$? I feel like I'm missing something here. Sorry if I put the title wrong..
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0answers
27 views

Prove whether a particular function is concave

Given the following equation: $$V(w) = - \frac{\alpha}{2} \left[ y_1(w) + y_2(w) + \int _{-\infty}^{+\infty} \vert y_1(w) - y_2(w) - x\vert f_{T1}(x)dx\right] \\- \beta \int _{w - y_1(w)} ^{+\infty} ...
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0answers
26 views

Convert time derivative to a function of time

Physics: I am asking for help to derive a general expression for the total amount of energy lost as a function of time from a radiating object. I'll simplify my problem like this: Say for example ...
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3answers
31 views

L'Hospital's Rule to find limit

I am asked to find $\lim\limits_{x \rightarrow \infty} {\left(\frac{8x}{8x+4}\right)}^{5x}$. Could anyone help me with figuring out how to start this problem? Thanks!
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0answers
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Real Analysis Differential functions

I am currently working through an exercise set and I am a little stuck on the following question: For $a > 0$, define a function $f_a(x)= \begin{cases} x^a \sin(1/x), &x \ne 0\\0, &x=0 ...
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0answers
13 views

Application of derivatives using particle movements

I have the values with me. But I am not quite sure how to use them. I pulled out a graph but it still made no sense to me. In question B, what do they mean exactly? A particle moves along a straight ...
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4answers
24 views

derivative of $\ln((1+\beta)^x-1)$

How do I differentiate the term $\ln((1+\beta)^x-1)$ with respect to $x$? Is it possible to do it this way: $$\frac{1}{(1+\beta)^x-1}$$ But i get stuck if i do the normal differentiation.
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1answer
24 views

Differentiating both sides of an equality with respect to first variables? (Not answered)

I am proving a statement and the truth of the following proposition would help me with it. If anyone could say whether the proposition is true and give a hint how to prove it I would be very much ...
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0answers
25 views

Elementary Analysis, 3rd root question

Prove that $\forall a \in \mathbb{R}$ there is a unique solution to $x^3 = a$ Prove that $\forall x,a \in \mathbb{R}$ $$(x^{1/3}-a^{1/3})(x^{1/3})^2 + a^{1/3} x^{1/3} + ((a^{1/3})^2)=x-a$$ Prove ...
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2answers
25 views

Use the definition of derivative to prove ln(x+1)/x =1

How would I use the definition of derivative to prove lim (as x->0) [ln(1+x)]/x = 1? I got to [ln(1+x+h)/(x+h) - ln(1+x)/x]/h but have no idea where to go from here. On another site I found ...
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1answer
18 views

Which of the following options are correct?

$f(x),g(x)$ are defined on $[-1,1]$, $f'(0),g'(0)$ exist, $f(0)=g(0)$, and $f(x)\ge g(x)$ holds for an open interval containing $0$. Then which of the following is correct: I, $f(x)$ and $g(x)$ have ...
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2answers
24 views

Why does gradient descent make sense?

Suppose I define two functions of $x$ in terms of a convex function $f$ with a unique minimum $x_0$: $$f_1(x) = 1 \times f(x)$$ $$f_2(x) = 2 \times f(x)$$ Suppose I wanted to minimize each of these ...
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0answers
13 views

Matrix derivative involving inverse

Suppose we have a symmetric matric $\boldsymbol{A}$ and its inverse is given by $\boldsymbol{A}^{-1}$. Then, how can we compute the derivative $$ ...
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ODE - Laplace transform

I have an ODE $\psi^{'}(s)_{3 \times 3}=(A+Bs)_{3 \times 3}\psi(s)_{3 \times 3} \tag1$ where A,B are constant skew symmetric matrices with zero determinant. $\psi(s)$ is a rotation matrix. It implies ...
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2answers
22 views

Derivative limits exercise [on hold]

Let $f: \mathbb{R} \to \mathbb{R}$ which is differentiable for every $x\in \mathbb{R}$. Prove that $$\lim_{h \to 1} \frac{f(hx) - 2f(x) + f(\frac{x}{h})} {h-1}= 0, \quad x \neq 0$$
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1answer
33 views

Application of the mean value theorem for Integrals

Suppose that $f(x)$ is a differentiable function in $[a,b]$, $f^{'}(x)$ is a monotone decreasing function in $(a,b)$, and $f^{'}(b)>0$. So how to prove that $$ \big \vert \int_a^b \cos ...
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2answers
30 views

derivative of $\frac{2}{3}x^{3-e}$

Find the derivative:$\;\;\;\;\;\;\dfrac{2}{3}x^{3-e}$ I am not sure how to solve this problem. My try: $\ln y=\dfrac{2}{3}(3-e)\ln x$ $\dfrac{1}{y}\times y\;'=\dfrac{2}{3}(3-e)\dfrac{1}{x}$ ...
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2answers
115 views

Differentiating with respect to the limit of integration

I'm confused about problems involving differentiation with respect to the limit of an integral, I just want to check that my understanding is correct. For example, are the following statements ...
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2answers
30 views

Making a piecewise function continuous and differentiable at point

Problem: Let $f(x) = \left\{ \begin{array}{lr} \frac{\arctan(x)}{(1+x)^2} & : x \geq 0\\ Ae^x + B & : x < 0 \end{array} \right. $ Find $A$ and $B$ such that the function is ...
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2answers
13 views

Critical points for undefined fraction on closed interval

I am told to find the absolute extrema of $$h(x) = \frac{8+x}{8-x},[4,6]$$ So I obtain the derivative of $$\frac{16}{(8-x)^2}$$ The trouble I am having is trying to determine the critical points. ...
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2answers
32 views

derivative of $e^{\ln x^2}-3x^7$

$$e^{\ln x^2}-3x^7$$ The first term: $=e^v$ $v=\ln x^2=u^2$ $v\;'=2uu\;'=(2\ln x)\dfrac{1}{x}=\dfrac{2\ln x}{x}$ $\dfrac{e^{\ln x^2}2\ln x}{x} +21x^{-8}$ How do I simplify further? I don't ...
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2answers
18 views

derivative of $y=\sqrt{10^{5-x}}=u^{1/2}$

$y=\sqrt{10^{5-x}}=u^{1/2}$ $y\;'=\dfrac{1}{2}u^{-1/2}\times u\;'=\dfrac{1}{2}(10^{5-x})^{-1/2}=\dfrac{1}{2\sqrt{10^{5-x}}}\times 10^{5-x}\ln10(-1)$ ...
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2answers
33 views

Evaluating $(\frac{\cos x}{1-\sin x})^2$

$(\dfrac{\cos x}{1-\sin x})^2$ $f\;'(x)= 2(\dfrac{\cos x}{1-\sin x}) \times (\dfrac{-\sin x+\sin^2x-\cos^2x}{(1-\sin x)^2})$ Does $\sin^2x-\cos^2=1$? or $-1$? Then it could factor with the ...
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2answers
36 views

Derivative of integral?

When asked questions of the type; What is the derivative of $f(x) = \int_0^{x^2} \frac{cos(t)}{t+1}dt $ ... what is the general method to solve them? Above is just an example from my workbook. I ...
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2answers
31 views

How to find the derivative of $f(x)=(x^3-4x+6)\ln(x^4-6x^2+9)$?

Find the derivative of the following: $$f(x)=(x^3-4x+6)\ln(x^4-6x^2+9)$$ Would I use the chain rule and product rule? So far I have: $$\begin{align}g(x)=x^3-4x+6 \\g'(x)=2x^2-4\end{align}$$ would ...
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2answers
35 views

Derivative Help: $f(x) = x^3\,e^{5x-7}$

I need to find the derivative of the following function: $${\rm f}\left(\,x\,\right)= x^{3}{\rm e}^{5x - 7}$$ but I don't know where to start with this problem. Please help.
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A question about a change of variable

I have came across this question while trying to find the derivate of the inverse functioin. And I have found the following limit: $$ \lim_{y\to y_0} = \frac{1}{\frac{f(x) - f(x_0)}{x-x0}}$$ We also ...
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0answers
24 views

Understanding integration and substitution

I'm an undergraduate student in EE. I often see that when talking about voltages, curents ... being expressed like functions of some independed variable (time) and when calculating integrals people ...
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1answer
33 views

Find the equation of normal line to the graph $y=2(x-1)^3$

Find the equation of normal line to the graph $y=2(x-1)^3$ at the point where $x=\frac12$. So far, I found the derivative: $$\frac{dy}{dx}= 6(x-1)^2 $$ What to do next?
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1answer
10 views

Help with Implicit Differentiation: Finding an equation for a tangent to a given point on a curve

When working through a problem set containing Implicit Differentiation problems, I've found that I keep getting the wrong answer compared to the one listed at the back of my book. The problem is ...