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
27 views

Differentiating product and squares of logarithms

I need help differentiating. I am really confused how to solve with the $\ln x$ in the equation. Which of the logarithm rules do I need to use for this equation? $$y= 12x \ln x + 12x - 6x (\ln x)^2 + ...
0
votes
0answers
13 views

Conditions for functions to be independent of one of their variables

I'm working independently through Spivak's Calculus on Manifolds and I've come across a stumbling block with respect to two of his questions. The first question is 2.22. If $f:\mathbb ...
0
votes
1answer
31 views

chain-rule application

Consider $f:\mathbb{R}^3\to\mathbb{R},(x,y,z)\mapsto x+y+z$ and a differentiable function $g:\mathbb{R}^2\to \mathbb{R}$. What is correctly if I want to apply the chain rule, ...
0
votes
2answers
29 views

Applying the chain rule correctly to $f(x,g(x))$

Consider a function $f:\mathbb{R}^2\to\mathbb{R},\; (x,y)\mapsto f(x,y)$, $g$ and $f$ continuously differentiable, $g:\mathbb{R}\to\mathbb{R}$. How to apply the chain rule on $f(x,g(x))$ correctly? ...
3
votes
3answers
52 views

Let $f:\mathbb{R} \to \mathbb{R}$ be $C^3$. Show the equivalence: $f^{(k)}(0) = 0$ for $k=0,1,2 \iff \lim_{x\to 0} \frac{f(x)}{x^3}$ exists

Let $f:\mathbb{R} \to \mathbb{R}$ be $C^3$. Show the equivalence: $$f^{(k)}(0) = 0 \quad k=0,1,2 \iff \lim_{x\to 0} \frac{f(x)}{x^3} \text{, exists.}$$ Trying: Since $f \in C^3$, implies $f, f', ...
3
votes
0answers
65 views

Show that such an $f$ cannot exist

Suppose $f:\mathbb R^n\to\mathbb R$ is a scalar field, such that for a given vector $a\in\mathbb R^n$ and any $y\in\mathbb R^n-\{0\}$ we have, $f'(a;y)>0$. Show that such a function $f$ cannot ...
0
votes
1answer
43 views

Differentiability issue with this function

$f:D\to{R}$ $$f(x)=\frac{1}{x-2}e^{\left|x\right|}$$ Find the domain $D$ of the function and study whether the function is differentiable. Find the left and right derivatives in the points where the ...
-6
votes
2answers
40 views

How to derive ∫f'(x)[f(x)]dx [on hold]

I need to show how ∫f'(x)[f(x)]dx is derived using integration techniques. Any specific examples that I can use to derive this formula? Thanks! Q: "Using clear explanations in the form of words and ...
1
vote
3answers
54 views

Indeterminate form $0^0$ using L'hospitals rule when calculating $\lim_{x\to0^+} x^{\sin(x)}$

Given the question $$\lim_{x\to0^+} x^{\sin(x)}$$ I have deducted so far that this has the indeterminate form $0^0$ so I have taken the natural logarithm of both sides to give me: $$\lim_{x\to0^+} ...
5
votes
2answers
58 views

Relationship between $\sin(a+b)$ and derivative product rule?

I noticed this interesting correlation between the sine angle addition formula and the derivative product rule. The sine addition formula is $$\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$$ The ...
8
votes
0answers
70 views

What is the Exterior Derivative Trying to Do?

$\newcommand{\R}{\mathbf R}$ Consider a smooth function $f:\R^n\to\R$ and let $Df:\R^n\to \R^{n*}$ be the map which takes a point $\mathbf a\in R^n$ to the linear map $Df_{\mathbf a}:\R^n\to \R$. ...
0
votes
1answer
29 views

Edited: Implicit Differentiation of Life-History Function

I am trying to implicitly differentiate the following function: $$ \lambda = \exp \left[ \left( \alpha + \frac{s}{\lambda-s} \right)^{-1} \right] $$ Can someone help me with this?
1
vote
2answers
38 views

Solving for y' in a fraction

Given the equation $x+xy^2 = \tan^{-1}(x^2y)$ find $y'$. I have tried doing this but solving for $y'$ I need some help and would like your advice. Work so far... ...
3
votes
1answer
60 views

How do I show $\lim_{x\to\infty}f(x) = \lim_{x\to\infty} f '(x)=0$ if $\lim_{x\to\infty}f '(x)^2 + f(x)^3 = 0$? [duplicate]

$f(x)$ is a real valued function on the reals, and has a continuous derivative such that $$\lim_{x\to\infty} f'(x)^2 + f(x)^3 = 0.$$ How do i show that $$\lim_{x\to\infty} f(x) = \lim_{x\to\infty} ...
1
vote
1answer
16 views

Problem regarding polynomials and partial derivatives

Let $P:\mathbb{R}^n\rightarrow\mathbb{R}$ be the homogeneous polynomial of degree $k$: $$P(x)=\sum_{|a|=k}c_{\alpha}x^{\alpha}$$ How can I show: $\partial^{\beta}P(x)=\beta !c_{\beta}$ for all ...
0
votes
0answers
26 views

In the derivative of an integral, can I compare the size of the limit effect to the area effect?

I have minimal formal math training and sometimes encounter problems like this where I am not sure what relevant techniques are available to use. Thanks for any advice you can give. My integral looks ...
0
votes
0answers
22 views

Finite difference expressions for spherical coordinates

I have information in a spherical grid, that is, I have a value for each combination of $\rho$, $\psi$, $\theta$ ($\rho$ - distance from origin to center of grid cell, $\psi$ - elevation angle to ...
2
votes
4answers
73 views

Is this correct? $ {d \over dy} (1+xy)^y = (1+xy)^y \cdot (1+x \cdot \ln(1+xy))$

I know the formula $ {d \over dx} x^x = x^x \cdot( 1+ \ln x ) $, but is below evaluation correct? $ {d \over dy} (1+xy)^y = (1+xy)^y \cdot (1+x \cdot \ln(1+xy))$
1
vote
1answer
63 views

A mean value theorem involving two functions [duplicate]

Let $f,g:[a,b] \rightarrow \mathbb{R}$ be continuous in $[a,b]$ and differentiable in $(a,b)$. Prove that there is a point $c \in (a,b)$ such that: $$[f(b)-f(a)]g'(c) = [g(b)-g(a)]f'(c).$$ I ...
-2
votes
3answers
39 views

Differentiation with respect to a constant variable? [on hold]

Let $y=f(x)$. If we are trying to find $f^{\prime}(x)$ and we know that in the domain we are trying to find $f^{\prime}(x)$ in, $x$ is constant , then what is $f^{\prime}(x)$? Is it zero?
2
votes
1answer
62 views

What did I do wrong?

So, I have found the following problem. This problem is a multiple-choice one, and I have to pick the correct answer. The problem, gives a function $f:D \to R$, $$f(x)=\frac{xe^x}{e^x-a}$$ with $a$ ...
4
votes
5answers
121 views

Differentiate expression involving reciprocal of square roots.

I need to differentiate $$5\over 2+\sqrt{1+3x}$$ I can get the answer from Wolfram Alpha but I'm trying to understand the working. Do I use the chain rule? My calculus is at the basic level.
2
votes
1answer
30 views

Find all $n \in \mathbb N$ such that $g(x) = 100|x+1| - \sum_{k=1}^{n}|x^k+1|$ is differentiable $\forall x$

Find all $n \in \mathbb N$ such that $$g(x) = 100|x+1| - \sum_{k=1}^{n}|x^k+1|$$ is differentiable $\forall x$. It's my high school calculus problem. Is it possible to solve this problem in the high ...
2
votes
1answer
28 views

Example where partial derivatives commute but are not continuous.

I am looking for an example of a function $f:\mathbb R^2\to\mathbb R$ such that there is a point $x\in\mathbb R^2$ with the following properties: 1) All partial derivatives of second order exist in a ...
2
votes
2answers
22 views

Necessary condition for local maximum

Let $\Omega\subset \mathbb{R}^n$ open, bounded and let $f:\Omega\to\mathbb{R}$ be a $C^2$-function. I want to prove: Necessary for a interior maximum $x_0\in\Omega$ is that $D^2f(x_0)$ is negative ...
2
votes
1answer
43 views

Calculate the derivative

I'm asked to find the derivative of the following: $$ \sqrt[4]{x} + \sqrt[3]{3x} $$ I attempted to solve the problem and got the following result, but my book says I am wrong. $$ \frac 14x^{-\frac ...
0
votes
0answers
44 views

Derivative of indicator function and summation [on hold]

What is the derivative of $G = \sum_{i=1}^{n}\Big(\mathbb{1}\{i \geq x + k\} v(x) + \mathbb{1}\{i < x + k\} v(i)\Big)$ with respect to $x$? Note that $\mathbb{1}\{\cdot\}$ is the indicator function ...
4
votes
4answers
177 views

Is My Proof that $\pi^e < e^{\pi}$ Valid? [duplicate]

The other day, a math teacher at my college gave me a challenge problem: Prove that $$\pi^e < e^{\pi}$$ without using a calculator. The next day, I found a valid proof, but I used a log table ...
0
votes
2answers
43 views

Differentiate the Function: $g(u)=\ln\left(\frac{\ln\ u}{1+\ln\ (2u)}\right)$

$$g(u)=\ln\left(\frac{\ln\ u}{1+\ln\ (2u)}\right)$$ $$=\ln\ (\ln\ u)-\ln(1+\ln\ (2u))$$ This is the part where I get a little confused. Keep in mind I am using this formula $$\frac{d}{dx}[\ln ...
0
votes
2answers
39 views

Measure of curve smoothness

Could someone please give me the intuition behind using integral of squared second derivative as a measure of curve smoothness? I was thinking that since curvature measures how fast a curve changes, ...
1
vote
2answers
55 views

Differentiate the Function $ h(z)=\ln\sqrt{\frac{a^2-z^2}{a^2+z^2}}$

Differentiate the function $$h(z)=\ln\sqrt{\frac{a^2-z^2}{a^2+z^2}}$$ My try: $$h(z) = \frac{1}{2}\ln\left(a^2-z^2\right)-\frac{1}{2}\ln\left(a^2+z^2\right)$$ so $$h'(z) = ...
1
vote
2answers
21 views

Solving for x using two derivatives and algebra.

There are two things I don't understand about the following: " Set these derivatives equal to each other and solve the resulting equation. $2\sqrt3\cos(x) = 2\sin(x)$ $= \sqrt3 = \tan(x)$ (since ...
1
vote
3answers
40 views

Differentiating the exponent power series

We know that $$ e^x = \sum\limits_{n=0}^{\infty}\frac{x^n}{n!} $$ We know that the series is uniformly convergent everywhere, and therefore we can differentiate term by term, i.e $$ ...
1
vote
2answers
55 views

Practical use for negative $dt.$

I am writing a section of notes for Calculus 1 on related rates. In the section where I discuss differentials, I write that the quantity $dt$ must be nonnegative. I imagined the only reason it would ...
-1
votes
4answers
36 views

Differentiation question (quotient rule)

Find $f'(x)$ if: $$f(x)=\displaystyle {6x \over \sqrt{ 1+x^2}}$$ Ans: $\displaystyle {6 \over (1+x^2) \sqrt {1+x^2}} $ My problem is that after applying the quotient rule, i can't simplify it to ...
0
votes
1answer
19 views

Total derivative product rule

Definition: Let $U\in \mathbb{R}^n$ be an open set. Let $a\in U$ and $f:U\to \mathbb{R}^m$. We say that $f$ is total differentiable at $a$ if there exists a matrix $T\in \mathbb{R}^{m\times n}$ and a ...
3
votes
1answer
246 views

If the derivative is zero on [a, b] so the function is constant - using Heine-Borel?

I know the proof using MVT but I was wondering if it can be proofed using Heine-Borel Lemma, that "Every open cover of close interval has a finite subcover". (without compactness, simple as that). ...
1
vote
2answers
46 views

Is 'a' differentiable in f when f is a product of a differentiable and non-differentiable function?

Recently, I was studying differentiable and non-differentiable functions and I wondered whether this "conjecture" of mine is true: 1) "If $f(x)$ is a function that is the product of $g(x)$ and $h(x)$ ...
-5
votes
0answers
46 views

Need help for calculating two derivatives analytically [on hold]

During building a Jacobian matrix for a numerical simulation I need to calculate following two derivatives where I hesitate about the correct answer: $$\partial( \partial X/ \partial\theta)/\partial ...
0
votes
0answers
17 views

Estimating the derivative of the difference function from that of a function

Suppose that $f$ is a twice differentiable function in an interval $(N,2N)$. We write $f_1(n,h)=f(n+h)-f(n)$, i.e., $f_1$ is the difference function. Then, a proof I'm reading estimates that if ...
2
votes
3answers
110 views

Derivative of $(-1)^x$

I'm taking a summer calc 2 class and we're getting into alternating patterns. I was interested in seeing the graph of $(-1)^x$ so I typed it into my TI-84 for $y = (-1)^x$. Surprisingly, the graph is ...
0
votes
1answer
37 views

Continuously differentiable operator

I consider an operator $A:H^1_0\to H^1_0$ defined by $$Au(t)=\int_0^1 G(t,s) f(s,u(s))ds$$ where $$ G(t,s)=\begin{cases} t(1-s), &t\leq s\\s(1-t), &s\leq t.\end{cases}$$ I want to know what ...
2
votes
2answers
46 views

Why is 1 not a critical point for this function?

For the function $f(x) = \frac{x^2}{x-1}$, why is $1$ not a critical point, along with $0$ and $2$? Don't critical points include discontinuities?
0
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2answers
31 views

Solving the logistic equation [on hold]

Please help to solve the following logistic equation: $$y'=y \cdot (1-y), \ ∀(t>0 \land y(0)>0)$$
2
votes
4answers
56 views

Can we take out a constant while differentiating?

In the solved example above, rather than taking $a^2x^4$ together and differentiating $a^2 = 0$, we differentiated $x^4$ and took out $a^2$. Why? Couldn't we have differentiated $a^2$ and gotten the ...
1
vote
1answer
35 views

How to resolve total variation $F(u)=f(u)+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$

Given $u(x)=[u_1(x)..u_N(x)]$, $0 \le u_i(x) \le 1, \sum_i^N u_i(x)=1$ and the cost function is: $$F(u)=f(u(x))+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$$ where $u_i(x)$ is a value that indicate ...
5
votes
2answers
2k views

What is the $1469^\text{th}$ derivative of $x^{532}-5x^{37}-4$?

I'm doing some basic calculus exercises on higher derivatives. But I'm stuck at a problem. The question is to find the 1469th derivative of $f(x)=x^{532}-5x^{37}-4$. I've read something about using ...
0
votes
0answers
32 views

Derivation of energy function

Given the following energy function $E(d)$ (also found here on page 3): $$ E_d = \sum_{x,y \in \Omega} \left(d_{x,y} - \hat{d}_{x,y}\right)^2 + \lambda \sum_{x,y} \left( ...
2
votes
1answer
33 views

derivative of a recursive vector-valued function

I have a recursive vector-valued function $$\mathbf{y}(t)=\mathbf{W}\mathbf{y}(t-1).$$ To compute the derivative of $\mathbf{y}(t)$ with respect to $\mathbf{W}$, do I need to use the product rule? ...
1
vote
1answer
40 views

Verify $\frac {\partial B} {\partial T} =$ $\frac{c}{(e^\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}e^\frac{hf}{kT}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(e^\frac{hf}{kT}-1\right)}$$ The correct answer (I believe) ...