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

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99
votes
1answer
5k 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 ...
19
votes
4answers
10k 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 ...
6
votes
3answers
10k 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 ...
11
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 ...
4
votes
1answer
442 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 ...
1
vote
2answers
4k 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 $g′(0) = 0$.
10
votes
6answers
7k 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 ...
11
votes
5answers
498 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.
8
votes
2answers
389 views

Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$

Suppose that $f(x)$ is differentiable on $[0,1]$ and $f(0) = f(1) = 0$. It is also known that $|f''(x)| \le A$ for every $x \in (0,1)$. Prove that $|f'(x)| \le A/2$ for every $x \in [0,1]$. ...
2
votes
1answer
119 views

Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals A

I ran across this problem in my Analysis class and can't seem to come up with a good solution. Here's the question: Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals $A$. $f$ is ...
14
votes
1answer
578 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 $f'(x) = f^{-1}(x)$ ?
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 ...
6
votes
3answers
2k 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 ...
2
votes
6answers
299 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 ...
1
vote
3answers
82 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; ...
21
votes
9answers
6k views

Why do we require radians in calculus?

I think this is just something I've grown used to but can't remember any proof. When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does ...
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 ...
17
votes
3answers
1k views

Any ideas on how I can prove this expression?

I don't have a lot of places to turn because i am still in high school. So please bear with me as i had to create some notation. In order to understand my notation you must observe this identity for ...
4
votes
1answer
562 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} ...
7
votes
1answer
502 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
2answers
603 views

Continuous function with linear directional derivatives=>Total differentiability?

As in the title: If $f\colon\mathbb{R}^n\to\mathbb{R}$ is continuous in $x$ and has directional derivatives $\partial_vf(x)=L(v)\,\forall v\in\mathbb{R}^n$, where $L$ is linear, does this imply that ...
7
votes
5answers
208 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 ...
3
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 ...
3
votes
1answer
189 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
99 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 ...
-2
votes
2answers
86 views

Simplify $(\cos x)^{2^{x^{\cos x}}}$ with respect to $x$ & $pi$ [closed]

Simplify $(\cos x)^{2^{x^{\cos x}}}$ with respect to $x$ & $pi$... if $x > 0$ and $cos(x)$ $> 0$
7
votes
1answer
10k views

Understanding the relationship between differentiation and integration

I am trying to understand the relationship between differentiation and integration. Differentiation has been introduced to me by this diagram: Which displays that the derivative of a point $x$ on ...
7
votes
1answer
236 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^+} ...
3
votes
2answers
946 views

Looking for help with a proof that n-th derivative of $e^\frac{-1}{x^2} = 0$ for $x=0$.

Given the function $$ f(x) = \left\{\begin{array}{cc} e^{- \frac{1}{x^2}} & x \neq 0 \\ 0 & x = 0 \end{array}\right. $$ show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$. So I have to show ...
6
votes
1answer
988 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 ...
5
votes
3answers
180 views

Derivative of function with 2 variables

I've leart in Calculus 1 that the derivetive of $f(x)$ is: $$\lim_{h\to0} \frac{f(x+h) - f(x)}{h}$$. suppose $f(x,y)$ is a function with 2 variables, does $$f'(x,y) = \lim_{h\to0} \frac{f(x+h, y+h) ...
1
vote
1answer
95 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: ...
4
votes
4answers
326 views

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

The answer is (I think) $x^{\sqrt{x}-0.5} (1+0.5\ln(x))$, but how?
2
votes
2answers
783 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 ...
1
vote
3answers
191 views

Limits and derivatives - two questions

I was asked to find two limits. Let $f$ be differentiable function at $x=1$ and $f(1)>0$. $$\lim_{n \rightarrow \infty}\left(\frac{f\left(1+\frac{1}{n}\right )}{f(1)} \right)^{\frac{1}{n}}$$ ...
7
votes
1answer
208 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 ...
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 ...
10
votes
1answer
280 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 ...
4
votes
2answers
405 views

Closed form for $n$-th derivative of exponential

I need the closed-form for the $n$-th derivative ($n\geq0 $): $$\frac{\partial^n}{\partial x^n}\exp\left(-\frac{\pi^2a^2}{x}\right)$$ Thanks! By following the suggestion of Hermite polynomials: ...
14
votes
1answer
412 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 ...
14
votes
0answers
389 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
767 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
2answers
1k 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 ...
3
votes
2answers
1k views

Trapezoid rule error analysis

How can I prove that the max error of the trapezoid rule for the integral $\int_{a}^{b}{f(x)\, \mathrm{d}x} $ is: $$\Delta=-\frac{1}{12n^2}f''(c)(b-a)^3 \text{for } c \in (a,b) \ ?$$ I know that to ...
5
votes
1answer
1k 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
258 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
5answers
3k views

Derivative of Integral

I'm having a little trouble with the following problem: Calculate $F'(x)$: $F(x)=\int_{1}^{x^{2}}(t-\sin^{2}t) dt$ It says we have to use substitution but I don't see why the answer can't just be: ...
2
votes
1answer
140 views

Laplace's equation in rectangle geometry

Consider Laplace's equation in a rectangle with length and width of a and b respectively, with following boundary conditions: All the boundaries with $x < a/2$ have Drichlet boundary condition ...
1
vote
2answers
1k 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 ...