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

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83
votes
1answer
4k views

How discontinuous can a derivative be?

There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property. To my knowledge, not many ...
6
votes
3answers
7k views

Second derivative “formula derivation”

I've been trying to understand how the second order derivative "formula" works: $$\lim_{h\to0} \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}$$ So, the rate of change of the rate of change for an arbitrary ...
12
votes
4answers
6k views

Discontinuous derivative.

Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is ...
1
vote
2answers
3k views

Show that the function $g(x) = x^2 \sin(\frac{1}{x}) ,(g(0) = 0)$ is everywhere differentiable and that $g′(0) = 0$

Show that the function $g(x) = x^2 \sin\left(\frac{1}{x}\right) ,(g(0) = 0)$ is everywhere differentiable and that $g′(0) = 0$.
4
votes
1answer
381 views

Quotient of two smooth functions is smooth

Let $f:\mathbb R\to \mathbb R$ be a $C^\infty$-smooth function. Suppose that $f^{(k)}(0)=0$ for $k=0,\dots,n-1$. Prove that the function $g(x)=f(x)/x^n$ extends to a $C^\infty$-smooth function on ...
11
votes
5answers
390 views

Closed form for $n$th derivative of exponential of $f$

What is the closed form for: $$\frac{\partial^n}{\partial x^n}\exp(f(x))=\exp(f(x))\cdot[????]$$
5
votes
2answers
3k views

What is the formula for nth derivative of arcsin x, arctan x, sec x and tan x?

Are there formulae for the nth derivatives of the following functions? 1) arcsin x 2) arctan x 3) sec x, and 4) tan x Thanks.
9
votes
5answers
2k views

Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$

I was messing around with the definition of the derivative, trying to work out the formulas for the common functions using limits. I hit a roadblock, however, while trying to find the derivative of ...
11
votes
1answer
468 views

Inverse of a bijection f is equal to its derivative

Does there exist a differentiable bijection $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f'(x) = f^{-1}(x)$ ?
2
votes
6answers
285 views

Derivative of $x^2$

This seems too easy, but here's the question: $x^2$ is $x + x + ...+ x$ (with $x$ terms). Its derivative is $1 + 1 + ... + 1$ (also $x$ terms). So the derivative of $x^2$ seems to be $x$. And ...
25
votes
6answers
1k views

Uses of $\lim \limits_{h\to 0} \frac{f(x+h)-f(x-h)}{2h}$

I have been wondering whether the following limit is being used somehow, as a variation of the derivative: $$\lim_{h\to 0} \frac{f(x+h)-f(x-h)}{2h} .$$ Edit: I know that this limit is defined in ...
13
votes
4answers
4k views

If a function has a finite limit at infinity, does that imply its derivative goes to zero?

I've been thinking about this problem: Let $f: (a, +\infty) \to \mathbb{R}$ be a differentiable function such that $\lim\limits_{x \to +\infty} f(x) = L < \infty$. Then must it be the case that ...
4
votes
1answer
467 views

Partial derivative confusion.

I don't understand partial derivatives. Here's an example that nails down my confusion: Suppose we have some variables $x$, $p$, and $q$ with $p=x^2$ and $q=e^x$. Then $$\frac{\partial q}{\partial p} ...
3
votes
1answer
157 views

$\lim_{x\to c}f'(x)=L$ implies $f'(c)=L$

Let $f:[a,b]\to\mathbb{R}$ be differentiable on $[a,b]$ and let $c \in(a,b)$. Suppose that $\lim_{x\to c}f'(x)=L$ some $L \in\mathbb{R}$. Without using L'Hospital's Rule, prove that $f'(c)=L$. Hint: ...
1
vote
1answer
74 views

$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$

Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. I want to demonstrate that if $$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$$ then $f$ is linear. I know this is a well-known ...
6
votes
1answer
859 views

“Strong” derivative of a monotone function

It is well known that if a function $f\colon \mathbb{R} \to \mathbb{R}$ is monotone then $f'$ exists almost everywhere. Is it true that if $f$ is monotone then there exists (edit: I mean exists ...
2
votes
1answer
1k views

$f$ not differentiable at $(0,0)$ but all directional derivatives exist

Consider the function : $$f: \mathbb{R}^2 \rightarrow \mathbb{R} , (x,y) \mapsto \begin{cases} 0 & \text{for } (x,y)=(0,0) \\ \frac{x^3}{x^2+y^2} & \text{for } (x,y) \neq ...
1
vote
1answer
74 views

Frechet derivative of shift operator in $l_2$?

Let $x \in l_2$ and $J(x) = \sum_{k = 1}^{+\infty} x_k x_{k + 1}$. Find $DJ(u)$ and $D(DJ(u))$. Attempted solution Since $x \in l_2$, then $\sum_{k = 1}^{+\infty}x_k < \infty$. Another fact: ...
6
votes
5answers
184 views

The derivative of $e^x$ using the definition of derivative as a limit and the definition $e^x = \lim_{n\to\infty}(1+x/n)^n$, without L'Hôpital's rule

Let's define $$ e^x := \lim_{n\to\infty}\left(1+\frac{x} {n}\right)^n, \forall x\in\Bbb R $$ and $$ \frac{d} {dx} f(x) := \lim_{\Delta x\to0} \frac{f(x+\Delta x) - f(x)} {\Delta x} $$ Prove that ...
4
votes
4answers
309 views

Differentiation of $x^{\sqrt{x}}$, how?

The answer is (I think) $x^{\sqrt{x}-0.5} (1+0.5\ln(x))$, but how?
1
vote
3answers
168 views

Limits and derivatives - two questions

I was asked to find two limits. Let $f$ be differentiable function at $x=1$ and $f(1)>0$. $$\lim_{n \rightarrow \infty}\left(\frac{f\left(1+\frac{1}{n}\right )}{f(1)} \right)^{\frac{1}{n}}$$ ...
1
vote
3answers
57 views

Proving a function is constant, under certain conditions?

The problem: Assume $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies $|f(t) - f(x)| \leq |t - x|^2$ for all $t, x$. Prove $f$ is constant. I believe I have some intuition about why this is the case; ...
28
votes
5answers
2k views

Are there ways of finding the $n$-th derivative of a function without computing the $(n-1)$-th derivative?

Say we have a function $f(x)$ that is infinitely differentiable at some point. Is it possible to find $f^{(n)}(x)$ without having to find first $f^{(n-1)}(x)$? If so, does it take less effort than ...
5
votes
1answer
167 views

Are “most” continuous functions also differentiable?

Let $A$ be a nonempty open subset of $\mathbb{R}$. Consider a function $f : A \rightarrow \mathbb{R}$. Given that $f$ is continuous, what is the probability that it is differentiable? I suspect it ...
10
votes
1answer
218 views

Define second derivative ($f''$) without using first derivative ($f'$)

The question I'd like to ask is this: If $f''(0)$ exists, does $f'$ exist in a neighborhood of $0$? Of course, under the standard definition of $f''(0)$, we have already assumed that $f'$ exists ...
14
votes
1answer
260 views

Definition of the nth derivative? [First post]

If the definition of the derivative is $$ f^\prime(x) = \lim_{\Delta x \to 0} \dfrac{f(x+\Delta x) - f(x)}{\Delta x} $$ Would it make sense that the nth derivative would be (I know that the 'n' in ...
12
votes
0answers
339 views

Divergence of the Derivative of the Prime Counting Function

On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written $$ \pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1} $$ with $ \operatorname{R}(z) = ...
7
votes
3answers
646 views

Finding the $n$-th derivatives of $x^n \ln x$ and $\frac{\ln x}{x}$.

How can I prove the following identities: $$ \left( x^n \ln x \right)^{(n)}= n! \left(\ln x + 1 + \frac{1}{2} +\cdots +\frac{1}{n} \right), \quad x>0, \quad n\ge 1, \tag{a}$$ $$ \left( \frac{\ln ...
6
votes
1answer
193 views

$\alpha$-derivative (concept)

I found the following definition: Given an real number $\alpha$, we say that a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is $\alpha$-differentiable at $0$ if exists the limit: $$\lim_{t \to 0^+} ...
8
votes
6answers
5k views

Using the Limit definition to find the derivative of $e^x$

I was wondering how we could use the limit definition $$ \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ to find the derivative of $e^x$, I get to a point where I do not know how to simplify the ...
5
votes
1answer
353 views

Continuous right derivative implies differentiability

A book of mine says the following is true, and I am having some trouble proving it. (I've considered using the Lebesgue differentiation theorem and absolute continuity, as well as elementary analysis ...
5
votes
3answers
1k views

Solve $\sum nx^n$

I am trying to find a closed form solution for $\sum_{n\ge0} nx^n\text{, where }\lvert x \rvert<1$. This solution makes sense to me: $\sum_{n\ge0} x^n=(1-x)^{-1} \\ \frac{d}{d x} \sum_{n\ge0} x^n ...
5
votes
2answers
992 views

Proving Thomae's function is nowhere differentiable.

I am given the function $$f(x)=\begin{cases} 0 \text{ ; when } x \text{ is irrational} \\\frac 1 q \text{ ; for } x=\frac p q \text{ irreducible fraction}\end{cases}$$ Spivak proved that for $a\in ...
4
votes
2answers
153 views

Multivariable calculus - Implicit function theorem

we are given the function $F: \mathbb R^3 \to \mathbb R^2$, $F(x,y,z)=\begin{pmatrix} x+yz-z^3-1 \\ x^3-xz+y^3\end{pmatrix}$ Show that around $(1,-1,0)$ we can represent $x$ and $y$ as functions of ...
4
votes
1answer
600 views

If a derivative of a continuous function has a limit, must it agree with that limit?

Suppose we have a continuous function $f : \mathbb{R} \to \mathbb{R}$. Suppose also that for a certain point $c$, $\lim_{x \to c} f'(x)$ exists. Must $f'(c)$ exist as well, and be equal to this limit? ...
4
votes
2answers
115 views

formal proof from calulus

Given $f:R \to R$, $f$ is differentiable on $R$ and $\lim_{x \to \infty}(f(x)-f(-x))=0$. I need to show that there is $x_0 \in R$ such that $f'(x_0)=0$ I am trying to prove it by contradiction .... ...
4
votes
3answers
227 views

Differentiate $\sin \sqrt{x^2+1} $with respect to $x$?

Differentiate $$ \sin \sqrt{x^2+1} $$ with respect to $x$? Can someone please help me with question, im very lost.
1
vote
1answer
63 views

There is a unique polynomial interpolating $f$ and its derivatives

I have questions on a similar topic here, here, and here, but this is a different question. It is well-known that a Hermite interpolation polynomial (where we sample the function and its derivatives ...
1
vote
1answer
90 views

A polynomial agreeing with a function and its derivatives

If we want $$p(x_i)=a_i, \qquad x_1 < \dotsb < x_{n+1},$$ then there is a unique polynomial of degree $\leq n$ that accomplishes this (Lagrange interpolation). If we want $$p(x_i)=a_i, \qquad ...
1
vote
2answers
851 views

Why does zero derivative imply a function is locally constant?

I've been trying to prove to myself that if $\Omega$ is an open connected set in $\mathbb{R}^n$, then if $f\colon\Omega\to\mathbb{R}^m$ is a differentiable function such that $f'(x)=0$ for all ...
1
vote
3answers
308 views

Application of derivative - helicopter problem

A helicopter of enemy is flying along the curve given by $y =x^2+7$. A soldier, placed at (3,7) wants to shoot down the helicopter when it is nearest to him. Find the nearest distance. Please guide ...
1
vote
2answers
538 views

What is the difference between $d$ and $\partial$?

After seeing the following equation in a lecture about tensor analysis, I became confused. $$ \frac{d\phi}{ds}=\frac{\partial \phi}{\partial x^m}\frac{dx^m}{ds} $$ What exactly is the difference ...
0
votes
2answers
117 views

Present a function with specific feature

Is there any function whose derivative at a point is positive but it is not ascending or whose derivative is negative but is not descending? I have thought about this a lot, but I cannot find ...
0
votes
2answers
266 views

A proof that $1=2$. May I know why it’s false? [duplicate]

im not good at formatting. May I know why it’s false $x^2= x+x+x+\ldots(x\; \text{times})$ apply derivative on both sides $=> \frac{d}{dx}(x^2)=\frac{d}{dx}(x+x+x+\ldots(x\; \text{times}))$ ...
0
votes
1answer
307 views

Evaluate the partial derivatives of the following function:

The function $f: \mathbb{R} \longrightarrow \mathbb{R}$ is defined by the rule $$f(x,y) = \begin{cases} \frac{x^5y}{x^4+y^2}, & (x,y) \neq (0,0), \\ 0, & (x,y)=(0,0). \end{cases}$$ ...
20
votes
1answer
467 views

$n^{th}$ derivative of a tetration function

I stumbled upon this very peculiar function last summer, namely: $f(x)=x^{x^{x^{...^{x}}}}$, where there is a number $n$ of $x$'s in the exponent, I tried to find the derivative for the function and I ...
14
votes
3answers
1k views

Fractional Derivative Implications/Meaning?

I've recently been studying the concept of taking fractional derivatives and antiderivatives, and this question has come to mind: If a first derivative, in Cartesian coordinates, is representative of ...
17
votes
4answers
1k views

Differentiating $y=x^{2}$

I am reading in a book about differentiating, but I am confused with one of the steps he takes. We start with: $$ \begin{align} y &= x^{2} \\ y + \mathrm{d}y &= (x + \mathrm{d}x)^2 \\ y + ...
16
votes
1answer
476 views

“Converse” of Taylor's theorem

Let $f:(a,b)\to\mathbb R$. We know that for every $c\in(a,b)$ we can write $f(t)=\sum_{i=0}^k a_i(c)(t-c)^i+o\left((t-c)^k\right)$ and $\forall i$ $a_i(c)$ is continuous (with respect to $c$). Can we ...
7
votes
4answers
378 views

Nth derivative of $\tan^m x$

$m$ is positive integer, $n$ is non-negative integer. $$f_n(x)=\frac {d^n}{dx^n} (\tan ^m(x))$$ $P_n(x)=f_n(\arctan(x))$ I would like to find the polynomials that are defined as above ...