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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2answers
10 views

Given the integral of an equation over one set of bounds find the integral over another set of bounds.

If $\int_{1}^{3}f(w)dw=7$, find the value of $\int_{1}^{2}f(5-2x)dx=7$ I think this problem has something to do with the fact that (5-2(2)) = 1 and (5-2(1)) = 3 and these are the bound of the ...
4
votes
4answers
120 views

What does $d\log\left(\frac{y}{x}\right)$ mean mathematically?

I am used to seeing derivatives written as $$\frac{df}{dx}.$$ But my economics professor keeps using notation like $$ d\log\left(\frac{y}{x}\right)$$ and I have no idea what this means. What does ...
0
votes
0answers
34 views

Transformation of a Partial Differential Equation

How can we convert $$\frac{\partial c}{\partial t} = M\left[\frac{\partial}{\partial x}\left(c\frac{\partial c}{\partial x}\right)+\frac{\partial }{\partial y}\left(c\frac{\partial c}{\partial ...
1
vote
1answer
22 views

Does $o(|x-a|^n)$ approximation by a polynomial imply existence of derivatives?

While reviewing the topic of Taylor expansion, I've noticed that while in all statements about the $n$th order Taylor polynomial of $f:\mathbb R \to \mathbb R $, it's always assumed that $f\in C^n$, ...
5
votes
2answers
183 views

The composition of a nowhere-differentiable function with a differentiable function.

This is actually Problem $ 17 $ from Chapter $ 10 $ of the Fourth Edition of Michael Spivak’s Calculus. The statement is quite simple, but I have not had any success in finding an example. Here is the ...
2
votes
1answer
63 views

Why are the Cauchy-Riemann equations in polar form 'obvious'?

In my book on complex analysis I'm asked to prove the Cauchy-Riemann equations in polar form, which I did. However, at the end of the question the author asks why these relations are 'almost obvious'. ...
0
votes
2answers
34 views

Differentiate $\sin^{-1}\left(\frac {\sin x + \cos x}{\sqrt{2}}\right)$ with respect to $x$

Differentiate $$\sin^{-1}\left(\frac {\sin x + \cos x}{\sqrt{2}}\right)$$ with respect to $x$. I started like this: Consider $$\frac {\sin x + \cos x}{\sqrt{2}}$$, substitute $\cos x$ as $\sin ...
1
vote
0answers
23 views

Newton's method $f(x) = \frac{1}{2} x^8 + \frac{8}{5} x ^5+ 3 x +10$ near the point $x = 3.$

$\displaystyle f(x) = \frac{1}{2} x^8 + \frac{8}{5} x ^5+ 3 x +10$ near the point $x = 3.$ Use $x_1 = 3$ as the initial approximation. Find the next two approximations, $x_2$ and $x_3$, to four ...
1
vote
3answers
27 views

Question about derivatives and derivative rules

What are the differences and similarities between finding the derivative using the definition and between finding the derivative using the derivative rules? What are the differences between the ...
1
vote
2answers
52 views

Find acceleration when v(t) = 0

I am struggling with this... This was a question I got wrong on a test, and I obviously did not even know how to solve it, so any help is greatly appreciated! This is a simple velocity/acceleration ...
3
votes
0answers
79 views

I can't find the critical points for this function. I showed my work :)

So, I have to find Critical Points of $y=\frac{1}{(x^3-x)}$ I know the derivative. Derivative = $(3x^2-1)/(x^3-x)^2$ To find Critical Points I equal to $0$. $x=1/\sqrt3$ and $x=-1/\sqrt3 $ But ...
0
votes
1answer
31 views

Derivative Word Problem about Virus Spreading

I had this question on a practice sheet for our calculus unit, and I am kind of confused by the following question. At lunch one day, the flu rapidly starts infecting the students at the school. ...
107
votes
5answers
3k views

Why can't differentiability be generalized as nicely as continuity?

The question: Can we define differentiable functions between (some class of) sets, "without $\Bbb R$"* so that it Reduces to the traditional definition when desired? Has the same use in at least ...
2
votes
3answers
54 views

Differentiation method for evaluating $ \sum_{n=1}^\infty \frac{n^2}{3^n} $

I evaluated the following infinite sum (the original and broader question regarding this sum can be found at Evaluating $\sum_{n=1}^\infty \frac{n^2}{3^n} $). $$ \sum_{n=1}^\infty \frac{n^2}{3^n} $$ ...
2
votes
2answers
63 views

Taking a time derivative of a function of 3 variables.

I have a function of $3$ variables which are all functions of $t$. $$x = \frac{v_1t-y}{\sqrt{(v_2/\dot{x})^2 -1}} \tag 1 $$ In the equation $v_1,v_2$ are constant and $x$ and $y$ are both function ...
0
votes
2answers
32 views

differentiation of a surface

Consider the surface $\{\left(x,y,z\right)\in\mathbb{R}^3\mid z=f\left(x,y\right)\}$ with $f(x,y) = x^3 + 2xy + y$. Show, using the definition of differentiability, that $f$ is differentiable ...
1
vote
1answer
47 views

How differentiable is the function $g(x) = \sum_n 2^{-n} f(x-r_n)$ where $f(x)=x^2 \sin\frac1{x}$?

This is an auxiliary enquiry (something like it may well be already discussed on MSE, but I haven't found it) resulting from a feeling of unease provoked by the question of this post. Taking the ...
0
votes
0answers
52 views

How can we maximize the following functional?

$\max_{} \; \int_0^1 \left( -\frac{1}{2} \left( \lambda_1(1-t) - \int_t^1 \lambda_2(s) ds \right)^2 - 1.25 \lambda_2(t) \right)dt + \lambda_1$ s.t $\lambda_1\geq0$, and $\lambda_2(t) \geq 0$ for ...
2
votes
1answer
59 views

Can a function be differentiable while having a discontinuous derivative?

Recently I came across functions like $x^2\sin(1/x)$ and $x^3\sin(1/x)$ where the derivatives were discontinuous. Can there exist a function whose derivative is not conitnuous, and yet the function is ...
3
votes
1answer
141 views

mean value theorem sin(b) - sin(a)

It's too much hassle to post it here as latex, to so here's the screenshot. I don't understand why |cos(c)| = 1 Why 1? Why not $\frac {\sqrt{3}}{2}$? Why absolute value assumes the max value a ...
0
votes
6answers
81 views

What does $\frac{d^2 u}{dt^2}$ mean?

When it comes to taking a derivative, what does $\displaystyle \frac{d^2 u}{dt^2}$ mean ? Does it mean taking derivative of the function twice with respect to $t$. If yes, why is then $d^2 u$ squared? ...
0
votes
0answers
15 views

Compute the derivatives of an equation

I have an equation which is equal to: $(-c/2)ln(x) + (-c/2)tr(diag(B^TSB)x^{-1})$ Where $c$ is a constant, $tr$ represents the trace, $diag$ represents the diagonal. $B$, $S$ and $x$ are three ...
7
votes
1answer
357 views

Challenging recurrence relation problem

I am starting out with the following: $$ \frac{d^n}{dx^n}[g(x)^{f(x)}] = \sum_{c=0}^n g(x)^{f(x)-c}\lambda_{n,c}(x) $$ Therefore: $$ \frac{d^{n+1}}{dx^{n+1}}[g(x)^{f(x)}] = ...
0
votes
3answers
61 views

Finding the dy/dx of a complicated function

I need urgent help on this question. I have no clue how to solve it as it's very complicated to me. The question is the following: Given $y=\frac{2xy}{x^2 + y}$ find $\frac{dy}{dx}$.
1
vote
1answer
28 views

What is the Hessian matrix of $x\mapsto f(Ax+b)$?

Let $A\in\mathbb{R}^{n\times n}$ and $b\in\mathbb{R}^n$ $f\in C^2(\mathbb{R}^n)$ and $\tilde{f}(x):=f(Ax+b)$ for $x\in\mathbb{R}^n$ It's easy to prove that $$\nabla\tilde{f}(x)=A^T\nabla f(x)$$ ...
2
votes
0answers
51 views

Differentiation Theorem

Assume that a function $f$ is integrable on $[a,b]$ w.r.t. an increasing function $g$, that $f$ is continuous at $c\in[a,b]$ and that $g$ is differentiable at $c$. Then the function defined by ...
0
votes
0answers
21 views

Does $f:[-5, 5] \rightarrow \mathbb{R}, \quad f(x) = x - 5 \cdot \arctan(2x + 1)$ have a local minimum or maximum at $-5$ or $5$?

Does $$f:[-5, 5] \rightarrow \mathbb{R}, \quad f(x) = x - 5 \cdot \arctan(2x + 1)$$ have a local minimum or maximum at $-5$ or $5$? I have discovered using the second derivative test that it has a ...
2
votes
1answer
41 views

Transforming integral equation to differential equation

I was given the task to find all continuous functions that satisfy the following equation: $$x \int_0^x {y }dx=(x+1) \int_0^x{xy}dx$$ I am quite new to differential equations so my first thought ...
0
votes
0answers
16 views

Curve torsion through $\mathbf{r}$

While learning torsion i came across formula $$\tau = \frac{\mathbf{r}'\mathbf{r}''\mathbf{r}'''} {\mathbf{r}''\cdot\mathbf{r}''} = ...
1
vote
2answers
32 views

How to find $ \frac{d (\tanh(kx))}{d x}=?$

I am tried to resolve the problem $$ \frac{d (\tanh(kx))}{d x}=?$$ where $k$ is positive value. I found one solution that is $$ \frac{d (\tanh(kx))}{d x}=\frac{k}{2\cosh^2(kx)}$$ Is it right? If ...
-1
votes
2answers
99 views

How to find dy/dx = - fx/fy?

I need some walkthrough in solving the following question: find dy/dx = - fx/fy? 3x^2 - y^2 + x^3 = 0. I need to know the method to solve this question. ...
1
vote
0answers
36 views

How to get 2nd partial derivative of a function of two vector variables

I am having trouble to calculate the expression: $$ \textbf{C}_{\textbf{q s}}\ \dot{\textbf{q}}\ \dot{\textbf{s}} = \frac{\partial^2 \textbf{C}}{\partial \textbf{q} \partial \textbf{s}}\ ...
9
votes
3answers
313 views

Finding a pair of functions with properties

I need to find a pair of functions $f$, $g$ such that $f$ is not differentiable at $x = 0$ $g$ is not differentiable at $f(0)$ $g \circ f$ is differentiable at $x = 0$ I've tried a lot of ...
2
votes
1answer
51 views

TI-84 gives 100 for d/dx(cube_root(x)) at x=0

My TI-84 Silver Edition is doing something strange. If $f(x)=\sqrt[3]{x}$, $\frac{d}{dx}\sqrt[3]{x}=\frac{1}{3\sqrt[3]{{x^2}}}$ At $x=0$, $\frac{d}{dx}f(0)$ is undefined. When I type ...
1
vote
1answer
22 views

Derivatives - optimization (minimum of a function)

For which points of $x^2 + y^2 = 25$ the sum of the distances to $(2, 0)$ and $(-2, 0)$ is minimum? Initially, I did $d = \sqrt{(x-2)^2 + y^2} + \sqrt{(x+2)^2 + y^2}$, and, by replacing $y^2 = 25 - ...
-1
votes
2answers
46 views

If $f(x)=\sin^2(3-x)$, then what is $f'(0)?$

I've been doing the math myself and my answer happened to be $-\sin(6)$, am I just being really stupid here and unable to convert it to any of the answers or my answer is wrong (or the answers are ...
1
vote
2answers
37 views

Is it true that $\frac{d}{dt}f(g(t),h(t))=f'(g(t),h(t))g'(t)+f'(g(t),h(t))h'(t)$

I want to solve the following question: We want to find $\frac{du}{dt}$ where $u(x,y)=x^2y^3$ and $x=1+\sqrt{t}$ and $y=1-\sqrt{t}$. I know we can just plug $x=1+\sqrt{t}$ and $y=1-\sqrt{t}$ in ...
3
votes
1answer
43 views

What is the formal name for the conformal laplacian?

\begin{align} L=R-4\dfrac{n-1}{n-2}\nabla^k\nabla_k \end{align} What is the formal name for $L$? I have seen it referred to as the conformal laplacian, however I thought I once read $L$ with a formal ...
3
votes
2answers
102 views

If $\lim\limits_{x \rightarrow \infty} f'(x)^2 + f^3(x) = 0$ , show that $\lim \limits_ {x\rightarrow \infty} f(x) = 0$ [duplicate]

If $f : \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function and $\lim\limits_{x \rightarrow \infty} f'(x)^2 + f^3(x) = 0$ , show that $\lim\limits_{x\rightarrow \infty} f(x) = 0$. I ...
3
votes
2answers
38 views

Find derivative of integrate square function [closed]

I am finding a solution of that function. Could you have me to resolve it $$F=\left( \int {(ax+b-c)}^2 dx \right) +\lambda_1(a-m)^2+\lambda_2(b-n)^2$$ where $c,m,n ,\lambda_1,\lambda_2$ are constant ...
2
votes
0answers
36 views

Derivative of inv: subset of linear automorphisms

I have no clue how to approach this problem, I've asked for some help from different people, but I have yet to comprehend it. The question is the following, Let $\mathcal L$($\mathbb C$$^n$) denote ...
1
vote
3answers
50 views

Power rule vs. Derivative rule

I have been learning about derivatives and need some answers. So the power rule is simple you just bring down a power such as $f(x)=x^2$ becomes $f'(x)=2x$. Then with the derivative rule we use the ...
0
votes
1answer
56 views

Find the derivative of $\arcsin(x)$ by just using the common rules

I need to find the derivative of $\arcsin(x)$ by just using the common rules of differentiation, such as sum, scalar multiplication, product, quotient rule, the chain rule and the inverse function. ...
2
votes
1answer
24 views

find the derivative of a function with natural log

find the derivative of $f(x)=\ln(x^4)(\sqrt{5x-3})$ I just need help getting to the answer. The first answer I got was $f(x)=(x^4)(2.5)+(5x-3)^{1/2}(4x^3)$.
-2
votes
1answer
45 views

Derivation with TI-nspire CX CAS not working [closed]

I try to get the partial derivative of: $$\frac{x^2y-xy^2}{2}$$ where $x \in \mathbb{R}$ with respect to $y$.. When I type it into my TI-nspire I get $xy$, however the answer should be ...
0
votes
1answer
29 views

Derivative of a function which is defined as a derivative

I'm new to this kind of stuff so maybe this is a stupid question but I don't even know what to search on the internet. My problem is that: find the derivative of the following function on $\Bbb R^3$ ...
0
votes
1answer
24 views

derivative calculation is more efficient than the integration calculation

I read that for $n$ sampled time points, the computation time required by the derivative calculation increases linearly with $n$, while the computation time required by the integral calculation is ...
0
votes
1answer
32 views

Why derivative of $(3x^2 - 2)^{\frac{2}{x}}$ can be written as $(3x^2 - 2)^{\frac{2}{x}} \cdot (ln(3x^2 - 2)\cdot\frac{2}{x})'$?

I am struggling with derivatives of exponents functions... Why derivative of $(3x^2 - 2)^{\frac{2}{x}}$ can be written as $(3x^2 - 2)^{\frac{2}{x}} \cdot (ln(3x^2 - 2)\cdot\frac{2}{x})'$? Where does ...
0
votes
3answers
36 views

Why derivative of $\sqrt[x]{x}$ can be written as $(\exp(\frac{1}{x}\log(x)))'$

I simply cannot understand why the derivative of $\sqrt[x]{x}$ can be written as $(\exp(\frac{1}{x}\log(x)))'$? Also, is that $\log$ the natural log or what?
0
votes
1answer
36 views

Find the number of divisors of $f'(1)$

The question is that: Let $f(x) = x^{25} + 2x^{24} + 3x^{23} + 4x^{22} + \cdots + 25x$. Find the number of positive divisors of $f'(1)$. How to find this number easily? Is there only one way: ...