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

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56 views

Finding the differential of $y=(u+1)/(u-1)$

I'm having trouble with differentials. I've been trying to learn about them online using great resources like PatrickJMT but I'm having trouble finding examples for this kind or problem. I hate asking ...
2
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0answers
59 views

Derivative changes sign for continuous and differentiable function

Give $f$ is continuous and differentiable, if $f'(a) < 0 < f'(b)$, can we say there exists a $c\in (a,b)$ such that $f'(c) = 0$ ? My gut feeling is yes, using Rolle's theorem. If $f(a) = ...
3
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2answers
96 views

Does a nondecreasing, differentiable function have continuous derivative?

Are the following statements true? How to prove or disprove? (1). Let $f$ be a nondecreasing, differentiable function on $[0,1]$. Then $f$ is absolutely continuous? To be stronger, (2). Let $f$ ...
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3answers
203 views

Understanding the logic behind particular derivative

I have $\frac{\partial (f(x) g(x))}{\partial x}$=$g(x) f'(x)+f(x) g'(x)$, I need to differentate this function with respect to x. $f(x)=(x+1) (x+2)^2 (x+3)^3 (x+4)^4$ However I do not see the ...
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2answers
39 views

Accuracy of linear approximations.

it's another day of calculus and I'm having trouble with linear approximations, perhaps you guys can help. I am unsure of how to calculate the 'accuracy' of these approximations, let me give you an ...
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2answers
86 views

$f$ is twice differentiable, $f + 2 f^{'} + f^{''} \geq 0$ , prove the following

Let $ f : [0,1] \rightarrow R$. $f$ is twice diff. and $f(0) = f(1) = 0$ If $f + 2 f^{'} + f^{''} \ge 0$ , prove that $f\le 0$ in the domain. Don't give complete solution, only hints.
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1answer
50 views

Finding derivative form the definition

I want to find the derivative of the function $f:\mathbb R^n\to \mathbb R^m$ at a point $x_0\in \mathbb R^n$, where $f(x)=c\in \mathbb R^m$, is a constant function. What I did is as follows: If $f$ ...
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26 views

Regarding methods of finding a derivative.

I read in the American Mathematical Monthly Descartes found away to calculate the slope of a tangent to a curve at a point specified. Called the Double tangent point method ( I think). This method ...
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30 views

A question about the differentiability of two Weyl sums

Consider the following functions, associated with certain trigonometrical sums: $$ f_{\alpha,\beta}(x) = \sum_{n=1}^{+\infty}\frac{\cos(n^{\alpha+\beta}x)}{n^{\alpha}},\qquad g_{\alpha,\beta}(x) = ...
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1answer
46 views

Finding the equation of more than one tangent line

I ran into a problem I have no idea how to begin, maybe you guys can help me out. I think maybe it has something to do with parametric equations? But this is just a guess. Find equations of both the ...
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2answers
51 views

Proving second derivatives

I'm asked to prove a theorem (if that is the right word) about double derivatives. I'm still struggling with understanding Leibniz notation and I could use a push in the right direction. It's easy ...
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3answers
107 views

Can you factor before finding derivative?

Say the function is $y=\frac{x^2-1}{x-1}$ Can you factor functions before finding the derivative or does that not work?
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3answers
56 views

Non integer derivative of $1/p(x)$

I need to find the $k$'th derivative of $1/p(x)$, where $p(x)$ is a polynomial and $k\in\mathbb{R}$ It dosen't have to be an explicit formula, an algorithm which finds a formula for some $k$ is fine. ...
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0answers
35 views

Maximize profit

my book (George F. Simmons - Calculus with analitic geometri) have the following question: An library could buy from the book publisher the book "Rituals" with a cost of 40.0 each. The manager from ...
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1answer
33 views

Is it possible to have a inflection on a vertical asymptote?

I found the derivative of a function to be f'(x)=8/x^3 and thus its second derivative as f''(x)=0/3x^2. After setting the second derivative to zero and doing the substitution into the parent function, ...
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3answers
103 views

Can an inflection exist if there's no max/min?

Very quick question: if a function doesn't have a maximum nor minimum, can it still have a point of inflection? I believe that these two go hand in hand and without one you can't have the other but ...
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0answers
53 views

How many continuous functions are differentiable? [duplicate]

Consider the set of continuous functions $\mathbb{R} \to \mathbb{R}$. I assume that the subset that are not everywhere differentiable accounts for almost all of them. Is this true? What is the precise ...
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3answers
76 views

Derivative of $\sqrt{x^2+1}$

Ive been given this rule and asked to differentiate $\sqrt{x^2+1}$, however I am not sure what I am missing.It is said that if f is differentiable at x and f(x)>0. ...
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2answers
25 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
76 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 ...
3
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1answer
54 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 $\lim\limits_{h\rightarrow 0}\frac{d(f(x),f(x+h))}{h}$ ...
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0answers
47 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
31 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
52 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 ...
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1answer
541 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 ...
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2answers
48 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.
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26 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
51 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
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3answers
58 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
44 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
26 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} { ...
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1answer
50 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
70 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 ...
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1answer
49 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
39 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
41 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
24 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
89 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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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 ...
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2answers
103 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
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1answer
28 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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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: $\sin (e^{x^2}) e^x$
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1answer
106 views

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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28 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
58 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
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) ...
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2answers
46 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
2k 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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1answer
18 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
106 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 ...