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

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3 views

Matricial differentiation $x x^{\top} b $

What is the drivative of $x x^{\top} b $ with respect to x, knowing that b is constant vector?
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6answers
41 views

Using Chain Rule and Product Rule to find derivative

I have to find the derivative of the following function: $$f(x) = (x^3+ 4)(4x^5 + 2x − 5)^{1/2}$$ To start solving this, I've dissected the equation and realize that I must use the product and chain ...
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1answer
27 views

Help in differentiation [on hold]

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 = ...
2
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0answers
23 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
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0answers
36 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 ...
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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 + ...
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1answer
42 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
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1answer
30 views

Total derivative proof [on hold]

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
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1answer
515 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
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1answer
59 views

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

From a derivative my professor said it's not yet simplified. how? i think this is it already.
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0answers
28 views

A question on limits

$$\lim_{h\rightarrow 0}\frac{2^{8\cos(h)}}{8h}\left [ sin^{8}(\pi/6+h))-sin^{8}(\pi/6) \right ]$$ MY ATTEMPT: for $\lim_{h\rightarrow ...
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1answer
12 views

Equation of a line with a positive gradient [on hold]

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.
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0answers
46 views

how do calculus this derivate? [on hold]

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

What and how do derivate? [on hold]

How do I derive this function? $f(x) = x(e^{-x^2})$ I need the first and second derivative.
0
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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 ...
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2answers
39 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
47 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 ...
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0answers
30 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 $n$, ...
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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
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1answer
24 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 ...
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3answers
64 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
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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}} - ...
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2answers
33 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 ...
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1answer
29 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 ...
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2answers
20 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 ...
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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 ...
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0answers
24 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
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2answers
101 views

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

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
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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 ...
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1answer
42 views

Math question: Calculus [on hold]

"A rancher would like to enclose two adjacent rectangular corrals that cover a total area of $12,000 ft^2$. If material for the fence costs $3.5$ usd per foot, find the dimentions (length and width) ...
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0answers
17 views

How to solve this problem about production and derivatives? [on hold]

If p(x) is equal to the production of a factor when there are x workers, then the average productivity of the work force is: A(x) = p(x)/x a) Find A´(x). Why does the factor need to hire more ...
2
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2answers
39 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: $\sin (e^{x^2}) e^x$
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0answers
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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?
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1answer
27 views

First derivative of this secial function

What is the derivative of the following function: $$f(x) = \frac{a}{((\sqrt{b+bx})(d-\sqrt{e+gx}))^2}$$
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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
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2answers
24 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) ...
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2answers
38 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 ...
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3answers
28 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: ...
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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
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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
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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 ...
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0answers
56 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
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1answer
12 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
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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
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0answers
36 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 ...
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3answers
192 views
+200

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 ...
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1answer
23 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!
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1answer
51 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 ...