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

learn more… | top users | synonyms (2)

1
vote
2answers
90 views

How to prove a function is not differentiable

Given: $$f(x) = \begin{cases} x^3 &&& x<2 \\x+6 && &x \geq 2 \end{cases} $$ I need to prove that $f(x)$ is not differentiable at $x=2$, what should I do? $$\lim_{x \to ...
0
votes
0answers
42 views

ODE with multiple simple conditions $f'(x)=f(x)(Ax+D ) $

I have an ODE to solve . The main issue is,in addition to solving it I have to keep some conditions too in the solution of f(x).. I am bit confused regarding how to deal with it. Equation is given ...
-1
votes
1answer
45 views

Find the rate of change of the balloon's radius, given the rate of change of its volume [closed]

So I have this balloon right? I am blowing air into it and it's volume is increasing at 4cm^3/s. I want to know what the rate of change will be when the balloon's radius is equal to 10cm (assuming ...
0
votes
1answer
35 views

how to differentiate an indicator function?

I'm reading this paper and I arrived at this part when they introduce a formula for what they call 'an indicator function'. Here is a shot: what I understood from the first two formulas is that I ...
1
vote
4answers
46 views

Question about tangent and slope

Given the graph of $y=-e^{-x}$ and that there is a tangent to the graph that crosses the x axis at $(-4,0)$ determine the slope of that line. So this seems like a simple question but I don't know why ...
0
votes
0answers
17 views

Linearizing non-linear least squares: Problem with derivatives

We want to approximate $$y_i \approx a b^{x_i}$$ and thus have $$S=\sum_{i=1}^m (ab^{x_i}-y_i)^2$$ as least squares error term. This term is not linear in b, so it is not easy to calculate its ...
2
votes
2answers
69 views

A function is real-differentiable iff it has a complex-differentiable extension

Is this conjecture true? A function $f:\Bbb R\to\Bbb R$ is real differentiable at $a$ if and only if there exists a complex-differentiable function $g:A\to\Bbb C$ for some neighborhood of $a\in ...
1
vote
1answer
35 views

Question about the Fundamental Theorem of Calculus

So I have studied the FOTC, but not really sure of what I read so this question is just to help me learn the FOTC and understand how to do problems like it. $$ if $$ $$F(x)=\int_0^x\sqrt{sin^3(t)}dt$$ ...
4
votes
2answers
57 views

How can you explain implicit differentiation?

So I am taking calculus 1 online from a local college (bad idea, but the only thing that fit my schedule). The professor used the notation $f'(x) =$ for EVERY function up until two weeks ago. All of ...
0
votes
2answers
37 views

Is there a continuous compact supported function $f: \mathbb{R}^n\rightarrow \mathbb{R}^{2n}$ such that $f^{-1}$ is continuous differentiable

Is there a continuous compact supported function $f: \mathbb{R}^n\rightarrow \mathbb{R}^{2n}$ such that $f^{-1}$ is continuous differentiable? I don't know which theorem is related to this question, ...
0
votes
1answer
31 views

Can you uniquely define a tangent line at a point for a 3D csurve?

Let f be a function of the form: $x=f_x(t); y=f_y(t);\text{ and }z=f_z(t)$. Does the derivative set of the 3 functions mean the tangent at a point on the curve of f? Thank you in advance.
4
votes
1answer
78 views

Why generalize the derivative for multivariable functions? [duplicate]

Sorry if this is a dupe (did a search, couldn't find anything). In single variable calculus, if the following limit exists: $$\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h},$$ then this expression ...
-1
votes
1answer
94 views

How to find the derivative of the function $ \int_{x}^{x^2}\frac{t}{\ln(t)}dt$? [closed]

The problem is to find $\displaystyle\frac{d}{dx}\int_{x}^{x^2}\frac{t}{\ln(t)}\,dt$ I could do this if I had the first clue on how to integrate $\dfrac{t}{\ln(t)}$ but even wolframalpha is giving ...
9
votes
2answers
288 views
0
votes
3answers
39 views

Finding the tangent line to the graph of $f(x)=(x+2)^{3/5}$ at $x=-2$

Does the graph of the function $f$ have tangent line at the given points? If yes, what is the tangent line? $f(x)=(x+2)^{3/5}$ at $x=-2$ solution: yes, $x=-2$ The derivative I found: ...
3
votes
4answers
73 views

Find the derivative of $y=x\sqrt{9-x}$

"Find the derivative of $y=x\sqrt{9-x}$." So this is what I have and now I'm stuck. \begin{align} y' &= x \frac{d}{dx}\left[(9-x)^{1/2}\right] + (9-x)^{1/2} \frac{d}{dx}(x)\\ &= x ...
2
votes
2answers
54 views

Show that the graph of $y=x^3\sin(\pi/x)$ extends to a smooth arc

Here's the problem: Let $y(x)$ be a real-valued function defined on the interval $x\in [0,1]$ by means of the equation $$y(x)= \left\{ \begin{array}{lr} x^3\sin(\frac{\pi}{x}) ...
1
vote
1answer
47 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 ...
0
votes
1answer
25 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
vote
2answers
60 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 ...
0
votes
1answer
11 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
40 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 ...
0
votes
2answers
31 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
vote
1answer
47 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 ...
-4
votes
0answers
20 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
0answers
28 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 ...
0
votes
3answers
99 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.
-2
votes
3answers
50 views

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

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
61 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 ...
1
vote
0answers
42 views

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

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 ...
2
votes
2answers
118 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 ...
0
votes
3answers
70 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
vote
0answers
32 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 ...
1
vote
2answers
86 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, ...
2
votes
2answers
49 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
votes
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) = ...
3
votes
2answers
80 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 $f$ ...
1
vote
3answers
169 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 ...
1
vote
2answers
27 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 ...
6
votes
2answers
82 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.
3
votes
1answer
44 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
votes
0answers
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 ...
2
votes
0answers
27 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
31 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
vote
2answers
39 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 ...
0
votes
3answers
70 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?
1
vote
3answers
53 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
vote
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
vote
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, ...
2
votes
3answers
89 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 ...