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

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3
votes
2answers
52 views

if $f(z),\overline {f(z)}$ are analytic then they are constant

I'm trying to prove this "theorem": if $f(z),\overline {f(z)}$ are analytic in some open set $\Omega \subseteq \mathbb C$, then $f(z)$ is a constant. Hint: Use Cauchy-Riemann equations to show that ...
0
votes
0answers
21 views

Show that $\Phi^*_t\mu=\mu \iff \mathrm{div}_{\mu}X=0$.

Let $\mu$ be a non-vanishing $1$-form on $\mathbb{R}^n$. Given a smooth vector field X on $\mathbb{R}^n$, we define the divergence of $X$ wrt $\mu$, denoted by $\mathrm{div}_{\mu}X$, by $L_X \mu = ...
1
vote
1answer
17 views

Sign of differentiable function near critical point

Background : We know that Brownian path oscillates infinitely often changing signs in any neighbourhood of $0$. I was trying to understand if this property holds because that Brownian paths are not ...
0
votes
1answer
26 views

Let $y = 2au,\;$ and $\;x=a(u)^2.\;\;$ Find $\,\frac{dy}{dx}\,$ [on hold]

Let $y = 2au,\;$ and $\;x=a(u)^2.\;\;$ Find $\,\dfrac{dy}{dx}\,$ at $u=\dfrac 1\pi$
4
votes
5answers
92 views

$f(0)=0$ and $\lvert\,f^\prime (x)\rvert\leq K\lvert\,f(x)\rvert,$ imply that $f\equiv 0$.

I'll start with the precise statement of the problem: Suppose that $f:[0,b]\to\mathbb{R}$ is differentiable, $\,f(0)=0$, and that there exists a real number $K\geq 0$ such that ...
2
votes
1answer
23 views

Power series for inverse of truncated power series of $e$

Let $T_n(x)=\sum_{k=0}^{n}\frac{x^k}{k!}$. I'm looking at the function $$f(x)=\frac{1}{T_n(x)}$$ and I would like to find the power series of this particular function. I know I can use Maclaurin ...
2
votes
1answer
30 views

derivative of implicit function

supposed I have a function which is $q_2f_2(δq_1+q_2)$ I want to know the second derivative of the function w.r.t $q_2$ Firstly, I took the first derivative w.r.t to $q_2$ and I got the result as ...
0
votes
1answer
39 views

Find a function $f$ on $\mathbb{R}^3$ such that $A^*df=0$

Let $A : \mathbb{R}^2 = \{(u,v)\} \rightarrow \mathbb{R}^3=\{(x,y,z)\}$ be given by $A(u,v)=(u,v,F(u,v))$ Find a function $f$ on $\mathbb{R}^3$ such that $A^*df=0$. I have tried ...
0
votes
1answer
32 views

Limit on a continuous differential equation

Let $f$ be a continuously differentiable function and let $L=\lim_{x\to\infty}(f(x)+f'(x))$ be finite. Does this imply that if $$\lim_{x\to\infty} f'(x)$$ exists, then it is equal to $0$?
5
votes
2answers
39 views

Sufficient conditions to have $f' = O(f(x)/x)$.

Suppose $f$ a nonnegative real-valued function, non-decreasing, $O(x^m)$ for some $m \in \mathbb{Z}_{\geqslant 0}$ and $C^1$, with $f'$ being monotonic and nonnegative. Are this sufficient conditions ...
2
votes
0answers
20 views

Number of terms in $f'$ as a polynomial of the number of terms in $f$

Take an elementary function $f$. The derivative $f'$ is an elementary function as well. Let $G(f)$ be the minimum number of terms required to express $f$ as a formula comprising a finite number of ...
7
votes
2answers
73 views

If f is differentiable on (a,b) continuous at a and f has bounded derivative must f be right differentiable at a?

If $f$ is differentiable on $(a,b)$ continuous at a, and $f$ has bounded derivative, must $f$ be right differentiable at $a$? In case answer to previous question is true, is the statement still ...
0
votes
0answers
27 views

Movement of Horse Position during a race

I am trying to determine how to trace a horses position in running during a race and sort them in order of the horses have the fastest foot speed. Here is a sample of the data: ...
1
vote
3answers
81 views

What's wrong with this derivative for $\left(1−\frac{1}{x}\right)^x$?

I asked a question about how to differentiate $(1−1/x)^x$ before, for $x>1$. The derivative I was told is ...
-1
votes
2answers
54 views

How do I take the 100th derivative of a polynomial [on hold]

How could I find $$f^{100}(x)$$ for $$f(x)=2x^{100}-7x^{80}+15x^{60}-27x^{40}-18x^{20}+300$$
-4
votes
2answers
44 views

Consider the function with its first and second derivative help please [duplicate]

$$f(x)=\frac{4x^2}{x^2+3} $$ $$f'(x)=\frac{24x}{(x^2+3)^2} $$ $$f''(x)=\frac{72(1-x^2)}{(x^2+3)^3}$$ a)What are the critical numbers(if any)? b)On what intervals is the function increasing and on ...
2
votes
2answers
51 views

20th derivative of a rational function

I could not find the 20th derivative of the function below : $$f(x) = \frac{2x}{x^2 - 4}$$ I have taken 1st and 2nd derivatives but I could not succeed at generalizing the derivative function.
0
votes
0answers
12 views

Multitangent to a polinomial function

I'm trying to build some exercises on tangents of functions for beginner students in mathematical analysis. In particular I would like to suggest the study of polynomial functions $ y = p (x) $ of ...
2
votes
1answer
46 views

Application of Rolle's Theorem and differentiation

Suppose $f: \mathbb{R}\rightarrow \mathbb{R}$ is differentiable with $f(0)=f(1)=0$ and $\{x:f'(x)=0\}\subset \{x:f(x)=0\}$. Show that $f(x)=0$ for all $x\in [0,1]$. My Work: By Rolle's Theorem ...
1
vote
3answers
56 views

proof of chain rule

Is my proof correct? show: $(g\circ f)'(x_0)=g'(y_0)f'(x_0)$ Since $f'(x_0) = \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}$ and Since $g'(y_0) = \lim_{y\to y_0} \frac{g(y)-g(y_0)}{y-y_0}$ Multiply ...
1
vote
1answer
36 views

How do I prove this derivation?

I hope you can help me with this one because I seem to not quiet get a start here :/ Lets say we got a $b\in\mathbb{R}_{\gt 0}$ and a $y\in\mathbb{R}$ and we define $b^y:=\exp\left(\ln b \cdot ...
2
votes
2answers
21 views

Show that $\lim_{n\to\infty}\sum_{k=1}^n\bigl|k\bigl(f\bigl(\frac{1}{k}\bigr)-f\bigl(-\frac{1}{k}\bigr)\bigr)-2f'(0)\bigr|$ exists

Suppose $f\in C^3[-1,1]$, show that $$\lim_{n\to\infty}\sum_{k=1}^n\left|k\left(f\left(\frac{1}{k}\right)-f\left(-\frac{1}{k}\right)\right)-2f'(0)\right|$$ exists. I realized that ...
2
votes
1answer
20 views

What is partial derivative of distance to line equation?

The distance from a point to a line is given by the equation: $$ \mbox{distance}\ = \frac{|ax + by +c|}{\sqrt{a^2 + b^2}}$$ What are the partial derivatives of this equation with respect to $a$, ...
0
votes
1answer
29 views

Checking where the complex derivative of a function exists

I have the following function: $$f(x+iy) = x^2+iy^2$$ My textbook says the function is only differentiable along the line $x = y$, can anyone please explain to me why this is so? What rules do we ...
1
vote
0answers
33 views

Looking for a partial converse of Rolle's theorem

Let $f: [a,b] \to \mathbb R $ be a continuous function differentiable in $(a,b)$ such that $f(b)=0$ and for some $c \in (a,b) , f'(c)=0$ ; then under what additional conditions can we conclude that ...
2
votes
1answer
57 views

Why this approach to differentiate $\log_{10}(x+1)^x$ does not work?

I am trying to differentiate $\log_{10}(x+1)^x$ but I don't get the correct answer, could you please help me? I know that one correct solution is the following: \begin{align} ...
0
votes
0answers
40 views

Finding $g$ such that $(f(g(x)))'=1$ when $f:[-1,1]\to S$

This is probably a simple question, but had a little trouble figuring it out, so hopefully someone here knows how to do it. Suppose $f:[-1,1]\to S$, where $S$ is some set endowed with a nonnegative ...
2
votes
1answer
25 views

Finding the PDF from the CDF where the CDF is not differentiable at some point

I got the following problem: Let $X$ be a continuous random variable with $CDF$ denoted $F_X$ defined as follows: $F_X(x)= \begin{cases} 1-x^{-4/3}, & x\in[1,\infty) \\ 0, & x\in ...
8
votes
3answers
138 views

How to solve the differential equation $(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$?

Solve $$(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$$ I tried the substitution $y^2=t$ ; $2y\:\text{dy}=\text{dt}$ to get $$(x^3)\:\text{dt}+(1-t)[(x^2+1)t-1]\:\text{dx}=0$$ ...
7
votes
0answers
78 views

Function's analytic continuation is its own derivative

This is the question we were asked at the university by our professor for complex analysis. Not as an exam, but as a challenge. I don't think he knew the answer himself. Find a nontrivial example of ...
-1
votes
0answers
58 views

What is the “actual definition” of the following?

Imagine you were standing on the ground and as the ground starts moving you stay exactly where you are. Your change in movement, by standing on a moving surface, is like a path-line. Suppose we take ...
1
vote
4answers
80 views

Differentiating $ \left( 1 - \frac {1}{x} \right)^x $

I have a calculus question. How does one differentiate $\left(1-\frac{1}{x}\right)^x$, for x>1? It should be positive right?
0
votes
2answers
46 views

Why is this map not surjective at the origin?

$f:\mathbb{R}^{n+1} \rightarrow \mathbb{R}$ $f(x) = |x|^2$ Then the derivative map is $df_{x}(v)=2\sum_{i}{x^iv^i}$ is surjective except at 0. Is it because at 0 df only goes to 0, and doesn't ...
-1
votes
1answer
32 views

Finding a differentiable inverse of $f(x)=x/\cos x$

Let $$ f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right) \rightarrow \mathbb{R} $$ be defined by $$ f(x) = \frac{x}{\cos x}. $$ We're supposed to show that $f$ has a differentiable inverse $$f^{(-1)}$$ ...
1
vote
0answers
29 views

Derivative of a composition of function - nice proof

Let's consider the well known "fake" proof below for the derivative of the composition of functions: Let $E,G$ be intervals of $\mathbb{R}$, let $F$ a subset of a normed vector space, let ...
3
votes
1answer
66 views

Derivability of a function with an infinity of zeroes

Let $F$ be a normed vector space and $a\in F$. Is there a non zero function $f:\mathbb{R}\rightarrow F$, such that $f'(a)=0$ and $f$ is $0$ an infinity of times in any neighborhood of $a$ ? If not, ...
1
vote
0answers
32 views

Differentiation of $\exp(A)$

Let's say we have $${\sigma(\exp(a\cdot X^{-1} \cdot a^\mathrm{T}))}/{\sigma X}$$ when I know that the term inside the exponent is essentially a scalar. Should I differentiate according to ...
1
vote
0answers
28 views

derivative of t distribution cdf wrt degrees of freedom

Given the cdf of a t distribution as follows: $T_\nu(x)=\frac{1}{2} + x\Gamma(\frac{\nu+1}{2}) + \frac{_2F_1 ...
3
votes
0answers
54 views

An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
0
votes
3answers
47 views

How to differentiate the function $f(x) = [ \frac{a+x}{b+x}]^{a+b+2x}$?

It has been given that, $$f(x) = \Big[ \frac{a+x}{b+x}\Big]^{a+b+2x}$$ How to prove , $$f'(0) = 2\ln \frac{a}{b}+ \frac{b^2-a^2}{ab}\Big[\frac{a}{b}\Big]^{a+b}$$ Do I have to take the logarithm of ...
0
votes
1answer
39 views

If $f$ s twice differentiable and satisfies the following constraints, prove $f'(0)>-\sqrt 2$

Let $f$ be a twice differentiable function on the open interval $(-1,1) $such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0 $and $f''(x) \le f(x)$, for all $ x\ge 0$. Show that ...
0
votes
2answers
63 views

Why doesn't $\ln (x)$ have an asymptote since its derivative is $1/x$?

My understanding is that the derivative gives the gradient of the function at that point. So for the function $x^2$, its gradient at point $x=10$ is equal to $20$. Extrapolating this to $\ln (x)$, ...
1
vote
2answers
35 views

Give an example of a function who is nondifferentiable on (0, 2) but has an antiderivative on (0, 2)

Originally when I was playing around with this problem, I tried to first find a function who was differentiable, but whose derivative was not differentiable at a specific point. So I figured out the ...
1
vote
2answers
62 views

How to find the derivative of $F(x)=\int_{x^2}^{4x^2} \sin \sqrt t\;\;dt$?

For a real number $t>0$, let $\sqrt t$ denote the positive square root of t. For a real number $x>0$, let $F(x)=\int_{x^2}^{4x^2} \sin \sqrt t\;\;dt$. If $F'$ is the derivative of $F$, then ...
2
votes
1answer
74 views

Functional Derivative (Gateaux variation) of functional with convolution

I have the following functional \begin{align*} F[f]=\int f(x) \log(g(x)) dx$ \end{align*} where $g(x)$ is given by convolution $g(x)=y(x) * f(x)=\int y(\tau) f(x-\tau) d\tau$, so \begin{align*} ...
1
vote
2answers
35 views

Matrix representation of the derivative of a smooth function

Let $V:\mathbb R^n\to\mathbb R$ be a smooth function and define the Hamiltonian function $H:\mathbb R^n\times\mathbb R^n\to\mathbb R$ (kinetic plus potential energy) by $$H(x,y):=\frac ...
0
votes
1answer
30 views

Express the limit in terms of $f'(x_{0})$

Find the following limit in terms of $f'(x_{0})$: $$ \lim_{h \to 0} \frac{f(x_{0} - 3h) - f(x_{0})} {h} $$ Any help would be appreciated.
3
votes
1answer
20 views

Chain rule notation for composite functions

Suppose I have a function $ f(x, y, g(x, y)) $ How would I express $ \frac{\partial f}{\partial x} $? Using the chain rule, you'd naturally come up with $ \frac{\partial f}{\partial x} + ...
2
votes
1answer
33 views

Derivatives defined on a discrete state space

Ive been looking at certain economic papers, and optimal control papers. They define a state variable, $\omega$, which follows a discrete time Markov Chain. Then they define a utility function ...
0
votes
3answers
25 views

Derivative with Logarithm Problem

I'm not sure how to approach this problem and solve it. $$y=\log_5\ln(x^3+6)^4$$