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
35 views

Help in differentiation [closed]

Can somebody please show the steps of how differentiation of Shannon's entropy yields the following result? $H = -\sum_{l=0}^{L-1} p(l)\log_2[p(l)]$ The result of differentiating is $H_m = ...
3
votes
1answer
36 views

Can all null-homotopy be made differentiable on arbitrary metric space?

Let $M$ be a metric, and assume that it is simply connected. For a closed curve $f$, we define it to be differentiable iff for any $x$ then $\lim\limits_{h\rightarrow 0}\frac{d(f(x),f(x+h))}{h}$ ...
1
vote
0answers
39 views

Taking derivative under the integral sign

Reading a textbook and stuck on this one detail... would like to confirm my understanding. The book defines a function $\eta \in C^1(\mathbb{R})$ satisfying $0 \leq \eta \leq 1$, $0 \leq \eta^\prime ...
1
vote
1answer
23 views

How is the power rule applied to whole numbers

For the following function, how does the $+1$ become $0$ when finding its derivative via the power rule? Original function: $f(x) = 6x^2 − 4x^{-1} + 5x^{-2} − 2x + 1$ Derivative: $f '(x) = 12x + ...
0
votes
1answer
44 views

Chain rule application in fundamental Theorem of Calculus

I have attached a question that I came across in trying to understand the fundamental theorem of calculus. The solution (highlighted with an arrow). I have difficulty understanding why the chain rule ...
0
votes
1answer
32 views

Total derivative proof [closed]

The wikipedia article does not prove it http://en.wikipedia.org/wiki/Total_derivative Neither the top articles in google search. Could somebody help me proving it? I've found this: ...
7
votes
1answer
520 views

Where is the error in my proof that all derivatives are continuous?

I know that this can not be true due to counter-examples but I don't know where the error in my reasoning is. Assumption: If $f(x)$ is differentiable in $\mathbb{R}$ then the derivative $f'(x)$ is ...
-3
votes
1answer
65 views

Simplify $\dfrac{3}{49 \sqrt[7]{x^4}} -5x^4$ [closed]

From a derivative my professor said it's not yet simplified. how? i think this is it already.
-1
votes
1answer
12 views

Equation of a line with a positive gradient [closed]

Two straight lines passing through the point (0,2) are tangent to the graph of the function y=1-x^2. Find the equation of the line with a positive gradient.
-4
votes
0answers
47 views

how do calculus this derivate? [closed]

how do calculus this derivative? $$f(x) = (2x-x^2)^{1/2}$$ And how do I calculus this derivative? $$F(x) = (\sin{x}/(1+\cos{x}))^2 $$
-1
votes
2answers
46 views

What and how do derivate? [closed]

How do I derive this function? $f(x) = x(e^{-x^2})$ I need the first and second derivative.
0
votes
0answers
23 views

How do you solve part (b) to this polynomial interpolation question?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the polynomial interpolation method. (b) Assuming $f(x)\in C^3$, evaluate ...
0
votes
2answers
40 views

How do you answer these questions regarding the Taylor series method?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the Taylor series method. (b) Assuming $f(x)\in C^3$, evaluate the ...
2
votes
3answers
50 views

How to evaluate $\lim_{x \to \infty}\left(1 + \frac{2}{x}\right)^{3x}$ using L'Hôpital's rule?

I'm stuck on how to evaluate the following using L'Hôpital's rule: $$\lim_{x \to \infty}\left(1 + \frac{2}{x}\right)^{3x}$$ This is a problem that I encountered on Khan Academy and I attempted to ...
0
votes
0answers
35 views

Differentiability of polynomials

Trivial question but I am confused with the notation If $p_{n-1}$ is a polynomial of degree $n-1$, is it $\in$ the differentiability class C^n$? Obviously if $p_n$ is a polynomial of degree $n$, ...
0
votes
1answer
22 views

What is the Jacobian of the following function

Consider a function F: $R^n \to R^n$ defined by $$f(u) = A*u*(n+1)+\lambda *B$$ Where A is a tridiagonal n-by-n matrix with -2 on the main diagonal and 1 on the off diagonals. B = $\begin{pmatrix} { ...
-1
votes
1answer
28 views

Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds. (Cf. discussion on p. 45.) Also note that though I were able to resolve the first problem the second one is still ...
0
votes
3answers
66 views

How to find the values of m and b?

How do I find the values of m and b when: a) the function is continuous in $x = \pi$ b) the function can be derivated in $x =\pi$ $$y=\begin{cases} \sin x & x<\pi \\ mx+b & x\ge ...
3
votes
1answer
43 views

Measuring sums of complex alternating series

Suppose we have functions $$f(x) = \sqrt{x}, \space g(f) = \frac{df}{dx}+\frac{d^2f}{dx^2}+\frac{d^3f}{dx^3}\space ...$$ Applying function f(x) to g(f) we get: $$g(f(x))=\frac{1}{2}x^{-\frac{1}{2}} - ...
1
vote
2answers
34 views

derivatives of a vector of functions with respect to a vector

Let $\vec W \in \mathbb R^3$. What is the general solution to: $$\frac{\partial}{\partial \vec{W}} \begin{pmatrix} f(\vec W) \\ g(\vec W) \end{pmatrix} $$ I think that in the ...
1
vote
1answer
30 views

Polynomial and its derivative have a common factor?

When is $gcd(p(x),p'(x))\ne 1$ where $p(x)$ is a polynomial? That is when does the derivative of a polynomial and the polynomial has a common factor? By when i mean some condition for the ...
0
votes
2answers
21 views

Global maximum and global minimum a combination of values

I have two variables $x$ and $y$. I can have them both in any combination of positive numbers that will add up to $1000$ and need to find the combination in which $z$ is at its minimum in the ...
1
vote
1answer
30 views

Derivative of logistic loss function

I am using logistic in classification task. The task equivalents with find $\omega, b$ to minimize loss function: That means we will take derivative of L with respect to $\omega$ and $b$ (assume y ...
1
vote
1answer
34 views

Please help me check this derivative work

I have $$ J_{\theta}(X) = - \frac 1 m \cdot \left[ y \cdot ln( h_{\theta} (X ) ) + ( 1 - y) \cdot ln ( 1 - h_{\theta}(X) ) \right] $$ I need $\frac d {d\theta} J_{\theta}(X)$. I tried many time, and ...
2
votes
2answers
102 views

Derivative of $(-2)^{x+1}$ [closed]

Can we compute the derivative of $(-2)^{x+1}$? This may sound silly, but think about it. We cannot apply any of our formulae on it. I think we may have to go old school with this one
0
votes
1answer
23 views

Sign convention for derivatives in a $\mathbb{Z}_2$ graded space

Suppose $V=V_0\oplus\theta V_1$ is a $\mathbb{Z}_2$ graded super vector space. Note: Since $\theta^2=0$, this implies $\theta\mathrm{d}\theta=-\mathrm{d}\theta\cdot\theta$. However, I wish to know ...
2
votes
3answers
65 views

How to find derivative of an integral of this type

$$f(x) = \int _x^{e^x}\:\left(\sin t^2\right)\,dt$$ How to find the derivative $f'(x)$ Attempt: $\sin (e^{x^2}) e^x$
4
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0answers
28 views
+100

Derivative of a generalized hypergeometric function

Let $$f(a)={_2F_3}\left(\begin{array}c1,\ 1\\\tfrac32,\ 1-a,\ 2+a\end{array}\middle|-\pi^2\right).$$ How to find $f'(0)$ in a closed form?
-1
votes
1answer
27 views

First derivative of this secial function

What is the derivative of the following function: $$f(x) = \frac{a}{((\sqrt{b+bx})(d-\sqrt{e+gx}))^2}$$
0
votes
3answers
54 views

Showing $\lim_{n \to \infty}\left(1 + \frac{x}{n}\right)^n = e^x$ using implicit and log differentiation

Hey guys I'm looking over my review sheet for an upcoming test and I'm having trouble with this problem. Apparently I'm supposed to use implicit differentiation and log differentiation, and I'm ...
0
votes
2answers
25 views

Find range of values

Find the range of values of the constant $a$ at which the equation $x^3 - 3a^2x + 2 = 0$ has $3$ different real number roots. I took the derivative and found that $x = -a, a$ Then I solved for $f(a) ...
1
vote
2answers
40 views

$\dfrac{\partial}{\partial x}\left(\int_{g(x)}^{h(x)}f(y)\, dy \right)= f(h(x))h'(x)-f(g(x))g'(x)$

I'm trying to prove the following, interesting, relation: $\dfrac{d}{dx}\left(\int_{g(x)}^{h(x)}f(y)\, dy \right)= f(h(x))h'(x)-f(g(x))g'(x)$ I tried to integrate by parts the RHS, but i don't ...
1
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3answers
37 views

maximum area of a rectangle inscribed in a semi - circle with radius r.

A rectangle is inscribed in a semi circle with radius $r$ with one of its sides at the diameter of the semi circle. Find the dimensions of the rectangle so that its area is a maximum. My Try: ...
-1
votes
2answers
49 views

Differentiability at x=0 [closed]

Discuss the differentiability of the following function in $x$ = $0$: $ f:\mathbb{R} \to \mathbb{R}: x\mapsto \begin{equation} f(x)= \begin{cases} \sqrt{x} & \text{if } x \geq 0 \\-\sqrt{-x} & ...
0
votes
1answer
17 views

Find the rate of change. $P=250(1+(2t/(49+t^2)))$

A population of bacteria is introduced into a culture. The number of bacteria $P$ can be modeled by $P=250(1+(2t/(49+t^2)))$ where $t$ is time (in hours). Find the rate of change of the population ...
5
votes
2answers
99 views

Is $\int^x \cos \frac1t$ differentiable at zero?

From Spivak's Calculus, 4th ed., exc 14-20: Let $$f(x) = \begin{cases} \cos \frac1x, & x\neq 0\\ 0, &x=0. \end{cases}$$ Is the function $\int_0^xf$ differentiable at zero? I'm having ...
1
vote
0answers
57 views

second order ODE :- solution

We have $y''-Py'-Qy = 0 $ where P,Q are $P = K_1+K_2x, Q =K_2 $. $K_1,K_2$ are constants. y' means derivative with respect to x . Please suggest a solution for y. Thanks
1
vote
1answer
13 views

Inverse Laplace transform, none factorable denominator

I am really stumpted on this problem and can't seem to figure out where to go from where I am. Can anyone give me some advice or hint where I should do next? Here is the problem: ...
0
votes
2answers
41 views

The fundamental difference that determines when a derivative can be calculated directly or only using the chain rule

I was given the following problem: Find $\frac{dy}{dx}$ using the implicit equation $x^2 + y^2 = 1$ What I'm more interested in is the explicit equation, $y = \sqrt{1 - x^2}$ (I'm allowed to ...
3
votes
0answers
37 views

Find all differentialbles function [closed]

Find all differentialbles function $f:[0,\infty)\rightarrow\mathbb R$ such that: a) $f^{\prime}$ is non-decreasing; b) $x^{2}f^{\prime}(x)=f^{2}(f(x)),~\forall x\in\lbrack0,\infty)$
4
votes
1answer
79 views

Derivative in 0

I'm a highschool student and we don't learn maths in English. So please excuse me for my Math's English. I'm doing an exercise and I can't answer its final question. Can you help me? Thank you! Let ...
8
votes
3answers
207 views

Determining the maximum value for the solution of this delay differential equation?

I am working on the following delay differential equation $$\frac{df}{dt}=f-f^3-\alpha f(t-\delta)\tag{1},$$ where $\frac{1}{2}\leq\alpha\leq 1$ and $\delta\geq 1$. I know that there are three ...
0
votes
1answer
24 views

Continuity of fixed points of an equation

Say I have a function $f(x,\theta)$ such that for each value of $\theta$ the function $f$ has a unique fixed point $x^*(\theta)$, i.e. $$x^{*}(\theta) = f(x^{*}(\theta),\theta)$$ My question is when ...
0
votes
1answer
45 views

Find $\frac{dy}{dx}$ of $y=\sqrt{u}$

Find $\dfrac{dy}{dx}$ of $y=\sqrt{u}$, $u=7-x^2$ This is on my homework and I don't know what to do exactly. Steps would be helpful!
-2
votes
1answer
53 views

Second derivative of this function [closed]

Whats is the second derivative of the following function? $$f(x) = \frac{0.12}{\sqrt{(0.01+0.24x)}\;(1.1-\sqrt{0.01+0.24x)}}$$ I get the first derivative as follow, but I am not sure is it correct ...
0
votes
1answer
26 views

Derivative from Graph

Hi, I am trying to study a bit ahead for my calculus class next year and I came across this question. I was wondering how to find the derivative of the graph without the function. I figured I could ...
0
votes
2answers
23 views

Determine the point(s), (if any), at which the graph of the function has a horizontal tangent.

Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. $y(x)= x^4-500x+2$ So I know the first thing to do is find the derivative which is: $y'(x) = 4x^3-500$ ...
1
vote
1answer
34 views

Having derivative in some $x_{0}$ implies having it in $U(x_{0})$ [closed]

Let $f$ be continious function in R and it has a derivative in $x_{0}$. Does it have derivative in some $U(x_{0})$?
3
votes
2answers
51 views

The right procedure on difficult related rates problems

I'm pretty sure the sample problems my teacher gives to us violate some article of the Geneva convention. I'm in talks with my embassy about that, but in the mean time maybe you guys could look over ...
1
vote
1answer
42 views

Why does secant method converge

Assume $f$ is continuous and twice differentiable on $[a,b]$ such that $f'(x)>0$ and $f''(x)>0$, $x \in [a,b]$. If $f(b)>0$ and $f(a)<0$ and I choose $x_0=a$,why are we gauraunteed ...