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

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1answer
56 views

Using the Fundamental Theorem of Calculus to find the derivative of an integral with variable lower limit

Let $f$ be a continuous real-valued function on $[a,b]$, and define $H$ on $[a,b]$ by $H(x)= \int_x^b f$. Find $H'(x)$. I use the Fundamental Theorem of Calculus $\int_a^xf(t)dt=F(x)$, but I am lost, ...
0
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1answer
8 views

Left & Right Area Approximation Using Y-Axis - Method Alternatives

Is there a simpler way of solving this then calculating x1(h)+x2(h)+x3(h)+x4(h) by using the given y values (in this case h, the height is one, because the length of each rectangle is one) ...
1
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1answer
41 views

Evaluate Derivative Using $\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ Definition

Evaluate the derivative of $x^3 - 3x +1$ using the $\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ definition to find the tangent of the curve at the point $(2, 3)$. I already calculated this derivative ...
2
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4answers
60 views

Derivative of function raised to a power, using the chain rule

How do I find the derivative of the function $f(x)= (2x+1)^2$? I've tried doing this problem and am not fully sure that I am correct. I found the derivative to be $f'(x) = 8x+4$. Is that correct?
1
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2answers
74 views

Derivative of matrix product: is it true that $\frac{d}{dt}(A^TA) = 2A^T \frac{dA}{dt}$?

$A$ is a square matrix. All elements of $A$ depend on a parameter $t$, that is, $a_{ij}=a_{ij}(t)$. Let $S(A):=A^TA$, and take the derivative of $S$ w.r.t. $t$: $\displaystyle \frac{dS}{dt}$ Now, ...
1
vote
2answers
24 views

Derivative of the trace of $X^TP^TPX$ with respect to P

$\newcommand{\Tr}{\operatorname{Tr}}$ Consider the following expression: $\Tr(X^TP^TPX)$ where $X$ and $P$ are real matrices. What is the best way to approach the calculation of its derivative ...
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1answer
8 views

Related Rates of Change - Cylinder Question

A cylindrical tank with radius 5 cm is being filled with water at rate of 3 cm^3 per min. how fast is the height of the water increasing? I dont want this question solved, but please help me correct ...
1
vote
1answer
36 views

Differentiable functions and examples

can someone give me an example of Differentiable function at x=4 and funcstions who dont Differentiable function at x=4? $f(x) = 2x-7$ $k(x) = 100x^7-55x^5+10000x^2$ $g(x) = 23$ Those are ...
-1
votes
1answer
84 views

The set of continuously differentiable functions such that $f(0)=0$, $f(1)=1$, $|f'|\le 3/4$

I came across the following problem: Let $C^{1}(\mathbb{R})$ be the collection of continuously differentiable functions on $\mathbb{R}$.Let $S$=$\{f \in ...
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2answers
29 views

derivative of this special function

I would like to take the first derivative of the following function respect to x. what is the derivative of this function with respect to ...
-1
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5answers
2k views

The rate of change of the distance from the plane to the radar station

Problem statement: A plane flying with constant speed of 4 km/min passes over a ground radar station at an altitude of 6 km and climbs at an angle of 35 degrees. At what rate, in km/min, is the ...
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 $\lim\limits_{h\rightarrow 0}\frac{d(f(x),f(x+h))}{h}$ ...
0
votes
2answers
53 views

what is the solution to ${\partial{\|A-RY\|^2_F}\over{\partial{Y}}}= 0$?

what is the solution to ${\partial{\|A-RY\|^2_F}\over\partial{Y}}= 0$? $A$ is $m$ by $n$, $R$ is $m$ by $s$ and $Y$ is $s$ by $n$ How to solve a quadratic equation? $AXA^T+B^TXB + X=C$, where X is ...
4
votes
0answers
27 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?
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1answer
39 views

L'Hopital's Rule with $\lim \limits_{x \to \infty}\frac{2^x}{e^\left(x^2\right)}$

(a) Show that $$\lim \limits_{x \to \infty}\frac{2^x}{e^\left(x^2\right)}$$ is a standard indeterminate form, but that L'Hopital's Rule does not give you any information about the limit. (b) Show ...
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votes
2answers
316 views

Finding the derivate of a function using first principles

I want to solve an equation from first principles. The first principles equation is: $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$ $$f(x) = \frac{1}{\sqrt{x}} \text{ at } x= 1$$ Basically, I ...
2
votes
2answers
106 views

Differentiability of the sum of the series $\sum_k \sin(kx)/k^2$

How to show the following: If $ f(x) = \displaystyle\sum_{k=1}^{\infty} \dfrac {\sin(kx)}{k^2} $, then show that $f(x)$ is differentiable on $(0,1)$ I guess it should be related to uniform ...
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: $\sin (e^{x^2}) e^x$
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votes
0answers
16 views

Optimisation: Maximum of a rectangle with semi circles at each end

A field is being built in the form of a rectangle with semi circles at each end. A $400$m racetract to is be built around the playing field. a) What Radius of the semicircular end would give the ...
0
votes
3answers
93 views

simplify the expression $\arctan\frac{x\sin t}{1-x\cos t}$

Same as above, how to simplify it. I am to calculate its $n$th derivative w.r.t x where t is const, but I can't simplify it. Any help would be appreciated. Thank you.
0
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0answers
27 views

If a continuous function $f$ on $[a, b]$ is differentiable and $f'\in L^1[a, b]$, can we conclude that $f$ is absolutely continuous?

If a continuous function $f$ on $[a, b]$ is differentiable and $f'\in L^1[a, b]$, can we conclude that $f$ is absolutely continuous? At first I don't think we can prove $f\in AC[0,1]$, because there ...
2
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0answers
37 views

Solve the initial value problem 0f $x'=f(x),\quad x(0)=y$ [on hold]

Solve the initial value problem $$x'=f(x),\qquad x(0)=y$$ for $$f(x)=(x^2,x+x^{-1})^T$$ Denote the solution by $u(t,y)$ and compute $$Ф(t,y)=\frac{du}{dy}(t,y)$$ Compute the derivative $Df(x)$ for ...
0
votes
3answers
125 views

The speed of the top of a sliding ladder

A $5$m ladder is leaning against a wall. If the bottom of the ladder is pulled along the ground away from the wall at a constant rate of $0.4$m/s, how fast will the top of the ladder be moving down ...
-2
votes
3answers
45 views

Derivative of $f(x) = \frac{(6x^2 + 2)}{ (x^2 - 1)^3}$ [on hold]

I can't seem to get the correct answer. Can someone please help demonstrate how to find the derivative of: $$f(x) = \frac{(6x^2 + 2)}{ (x^2 - 1)^3}.$$ Thank you in advance!
3
votes
1answer
51 views

What's the Differential of this Map $f:S^3\rightarrow \mathbb{R}$

$f:S^3 \rightarrow \mathbb{R}$ is defined as $f(x,y,z,w)=x+zw$, where $S^3= \{(x,y,z,w) | x^2 +y^2 +z^2 +w^2 =1\}$ I tried using a stereographic chart but that got ugly. The function is so simple I ...
0
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3answers
67 views

Find the derivative of $\frac{x^{1/3}} {({x^3+1})^{1/3}}$

I tried to solve it my answer is $$\frac{-2x^{4/3}(x^{3}+1)^{2/3}+1}{3x(x^3+1)^2}$$ I just want to make sure if I derived it correctly thanks
1
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0answers
31 views

Minimize distance between the ships

I'm studying calculus from a the book "Calculus with Analytic Geometry by Georfe F Simmons", and I have a certain difficulty to solve the following problem: Ah noon the ship A is at a distance at ...
2
votes
2answers
47 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 ...
3
votes
2answers
76 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 $f$ ...
2
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0answers
54 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) = ...
1
vote
2answers
25 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 ...
1
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3answers
161 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 ...
3
votes
2answers
97 views

Differentiating $x+2x^2\sin(1/x)$ near $0$, discontinuity of the derivative

I have this function $$ \begin{array}{l} f:\mathbb{R}\rightarrow\mathbb{R}\\ x\rightarrow\left\{\begin{array}{ll} x+2x^2\sin\left(\frac{1}{x}\right)&x\neq 0\\ 0&x=0 \end{array}\right. ...
3
votes
1answer
41 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 $f$ ...
0
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3answers
63 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?
6
votes
2answers
70 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.
1
vote
2answers
54 views

Find the derivative of $y=\cos(x) - 2\sin(x),$ when the gradient is $1$

I need to find the smallest positive value of $x$ for which the gradient of the curve has value 1. For this equation: $$ y =\cos(x)-2\sin(x) $$ The answer is 2.5c grad. The following is my ...
3
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1answer
84 views

The monotonicity of function $y=x/2 + x^{2}\sin(1/x)$ near $x=0$

Is the function $y=x/2 + x^{2}\sin(1/x)$ monotonic near $0$? The derivative $f'$ obviously goes positive and negative near $0$, because $$f'(x)= \frac12 + 2x\sin(1/x) - \cos(1/x))$$ Does that mean ...
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0answers
25 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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0answers
23 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) = ...
1
vote
1answer
29 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 ...
1
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2answers
36 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 ...
2
votes
3answers
87 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 ...
3
votes
2answers
80 views

Number of real roots of the equation $2^x = 1+x^2$

Find the number of real roots of the equation $2^x = 1+x^2$ My try: Let we take $f(x) = 2^x-1-x^2$. Now for Drawing Graph of given function, we use Derivative Test. $f'(x) = 2^x \cdot \ln ...
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vote
3answers
50 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. ...
1
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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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0answers
30 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 ...
1
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1answer
30 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, ...
3
votes
0answers
52 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 ...
1
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3answers
65 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. ...