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

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0
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1answer
19 views

Limit of a summation, using integrals method

I have seen an interesting question on stackexchange, which I would like to requote so that I can understand the answer =) $\lim\limits_{n\to\infty} \dfrac{1^{99} + 2^{99} + \cdots + ...
0
votes
1answer
17 views

Continuity and Differentiability of a series of functions

Consider the function $f(x)=\sum_{n=1}^{\infty} 2^{-n}g(2^{2^{n}}x)$ where \begin{equation} g(x)=\begin{cases} 1+x &-2 \le x \le 0 \\ 1-x &0 \le x \le 2 \end{cases} \end{equation} where ...
1
vote
0answers
19 views

Finding the density function of a function of random variable $X$

For two distinct density functions $f_0$ and $f_1$, log likelihood ratio is given as $$l(x)=\log\left(\frac{f_1(x)}{f_0(x)}\right)$$ Let $f_0\sim\mathcal{N}(-0.9,1.1)$ and ...
3
votes
1answer
25 views

Find derivative of tricky logarithmic functions

Find the derivative of $y=(x^{x+1})(x+1)^x$ So this is what I have, $$\ln y=\ln[(x^{x+1})(x+1)^x]$$ $$= \ln x^{x+1} + \ln(x+1)^x$$ $$\frac{1}{y}y' = (1)(\ln x) + (x+1)\frac{1}{x} + (1)(\ln(x+1)) + ...
0
votes
2answers
27 views

find the equation of the tangent line to the curve y=x^4-6x and perpendicular to the line x-2y+6=0

this is all I got right now: $y=x^4-6x$ $y'=4x^3-6$ $x-2y+6=0$ $y=(1/2)x+3$
1
vote
1answer
36 views

Why in this derivation we have ${1\over{\cos y}}={1\over{(1-x^2)^{1/{2}}}}$?

Let $\sin^{-1}x=y$, then $\sin y = x $ and therefore: $$\eqalign{\cos y \,{dy\over{dx}} = 1&\Longrightarrow {dy\over{dx}} = {1\over{\cos y}}\\ &\Longrightarrow {dy\over{dx}} = ...
0
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1answer
29 views

Please help me understand the concept of variable, and differentiation of variables.

`I am in the first year of college and know mathematical analysis in a very rigorous context, from high school/ math olympiads Imo's etc. But the concept of $df$ seems totally weird and ...
0
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1answer
27 views

characteristic function differentiation

Let $\mu$ be a probability measure on $\mathbb{R}$. Then the characteristic function is: $$ \varphi: \mathbb{R} \rightarrow \mathbb{C} \;\;\ \varphi(t):=i\int_\mathbb{R} e^{itx}d\mu(x) $$ Prove with ...
-1
votes
2answers
25 views

Integral Differentiation over constants [on hold]

Let $f(x)$ be integrable on $x\in[0,X]$ and $a,b>0$ constants. I would like to get the derivative of $$I(x)=a\int^x_0{(b-X-f(x))dx}$$ with respect to $b$, i.e. $\dfrac{\partial I(x)}{\partial b}$. ...
1
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1answer
12 views

Domain of dericative of a function [on hold]

Give an example of a function the domain of whose derivative is a PROPER subset of its own domain.
1
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1answer
21 views

Multivariate Calculus - Partial Derivatives - Implicit Differentiation - Chain Rule

Let $z = z(x,y)$ be defined implicitly by $F(x, y, z(x,y)) = 0$, where $F$ is a given function of three variables. Prove that if $z(x,y)$ and $F$ are differentiable, then $$\frac{dz}{dx} = - ...
0
votes
2answers
24 views

Find the equation of the parabola given the tangent to a point and another point.

I have a problem with derivatives, I've been trying to solve but I was not able to do it. A parabola is tangent to the line $3x-y+6 = 0$ in the point $(0,6)$ and goes through the point $(1,0)$. ...
1
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1answer
21 views

Existence of function with prescribed values?

Does there exist an infinitely differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ equal to $|x|$ when $x \in \mathbb{Z}$?
0
votes
1answer
22 views

Chain rule for $\ln(-f(\pmb{x}))$

I am trying to figure out how to calculate, for a smooth function $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ the second and higher order derivatives of $\ln(-f(\pmb{x}))$. I am not sure the notation, ...
1
vote
0answers
17 views

Differences between directional derivatives

In our Calc 3 class, we have started doing directional derivative and their applications. So, for a function $f(x,y)$, the value of $f_{xx}f_{yy}-f_{xy}f_{yx}$ is used to determine what type of ...
2
votes
1answer
28 views

Differentiation of $f(x)=2x(2+3x^2)^3$

The question: Differentiate $f(x)=2x(2+3x^2)^3$. How do I approach this problem? Do I only have to use the product rule...? I have the answer but I don't know how to get there. Here is my ...
0
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3answers
25 views

Difference between two derivatives

What is the difference between these two equations, and their answers? I'm meant to find dy/dx, or just derivative of y. $y_1= \cos(5x+1)^9$ & $y_2= \cos^9(5x+1)$
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0answers
40 views

Matrix differenciation

Need help finding the derivative of $$ \begin{align*} y'.y - β^T.x^T.y - β.x.y^T + β^T.x^T.x.β + l|β| \end{align*} $$ with respect to β $$ \begin{align*} \frac{dG}{dβ}& y'.y - β^T.x^T.y - ...
0
votes
1answer
18 views

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 ...
5
votes
3answers
73 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 ...
3
votes
1answer
206 views

How to differentiate this negative power? [duplicate]

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 ...
0
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1answer
19 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
16 views

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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0answers
35 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 ...
0
votes
1answer
14 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 $a$. ...
0
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4answers
32 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 ...
4
votes
1answer
89 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 $n!$. ...
0
votes
1answer
11 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 ...
0
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2answers
35 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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0answers
13 views

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.
0
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0answers
16 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 ...
2
votes
3answers
48 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: $$\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$ ...
0
votes
1answer
21 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 ...
0
votes
1answer
14 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: ...
1
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0answers
18 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 ...
0
votes
2answers
29 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.
0
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1answer
19 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
30 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! :)
0
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2answers
53 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..
1
vote
1answer
43 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} ...
1
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0answers
29 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 ...
1
vote
3answers
33 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!
2
votes
0answers
21 views

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 ...
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 ...
3
votes
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.
0
votes
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 ...
0
votes
0answers
27 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 ...
1
vote
2answers
27 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 ...
0
votes
1answer
20 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 ...
1
vote
2answers
42 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 ...