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

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1k views

Are all continuous one one functions differentiable?

I was reading about one one functions and found out that they cannot have maxima or minima except at endpoints of domain. So their derivative , if it exists, must not change it sign , i.e. , the ...
12
votes
3answers
410 views

The closed form of $\lim_{x\to\frac{4}{3}}\frac{\partial}{\partial x}\left[\,_2{\rm{F}}_1\left(\frac{1}{3},1;x;-1\right)\right]$

Do you think the following limit might have a closed form? Some hints or clues? $$\lim_{x\to\frac{4}{3}}\frac{\partial}{\partial x}\left[\,_2{\rm{F}}_1\left(\frac{1}{3},1;x;-1\right)\right]$$
12
votes
2answers
346 views

Finding the derivative of $x\uparrow\uparrow n$

I am trying to find a general derivative for the function: $f(x)=x^{x^{x^{...^{x}}}}$however to do that I must find $f^{\prime }$ and $f^{\prime \prime}$...etc. I am now trying to write down a general ...
12
votes
3answers
300 views

Simplified form for $\frac{\operatorname d^n}{\operatorname dx^n}\left(\frac{x}{e^x-1}\right)$?

I have found the following formula: $$\frac{\operatorname d^n}{\operatorname ...
12
votes
2answers
562 views

In what sense is the derivative the “best” linear approximation?

I am familiar with the definition of the Frechet derivative and it's uniqueness if it exists. I would however like to know, how the derivative is the "best" linear approximation. What does this mean ...
12
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2answers
562 views

Derivative of $x^{x^{\cdot^{\cdot}}}$?

The infinite tetration is defined as $$f(x)=x^{x^{\cdot^{\cdot}}}$$ This function is defined for $e^{-e} \leq x \leq e^{e-1}$. (Wikipedia image) Can one determine the derivative of this function? ...
12
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2answers
119 views

Second derivative of $f(f(\cdots f(x)\cdots )?$

For convenience, let's write $f_n(x)=f(f(\cdots f(x)\cdots )$ where $f$ is iterated $n$ times. Suppose: $$f(0)=0,\quad f'(0)=\alpha,\quad f''(0)=\beta$$ What is $f''_n(0)?$ I've found ...
12
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1answer
230 views

Zeroes of derivatives of high order

The problem is following. Let $f:(-1,1)\to [-1,1]$ has $n$ derivatives. Prove that there exists a number $\alpha_n$ (independent from $f$) such that condition $|f'(0)|\geq \alpha_n$ implies that ...
12
votes
2answers
566 views

Zeroes of the third derivative of an iterated sine.

I've been playing with the functions $$f_n:[0,\pi/2]\to[0,1]\\\begin{cases} f_1&=&\sin\\f_{n+1}&=&\sin\circ f_n\end{cases}.$$ A simple argument proves that $f_n(x)\to 0$ for $x\in ...
11
votes
6answers
949 views

Evaluating the derivative of $\large \;e^{e^x}$?

I know that the derivative of $\,e^x\,$ is $\,e^x$. But how do I evaluate $\dfrac{d}{dx}{\large\left(e^{e^x}\right)}\,$?
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 ...
11
votes
4answers
253 views

Why does differentiating a polynomial reduce its degree by $1$?

This may seem a bit silly but I am wondering: can it intuitively be shown that the derivative of a polynomial is precisely 1 degree lower than itself? I understand the basics of calculus enough to ...
11
votes
2answers
303 views

Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?

Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable? Similarly, let $f :\mathbb{C}→ \mathbb{C}$ be a function such that $f^2$ and $f^3$ ...
11
votes
5answers
497 views

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

What is the closed form for: $$\frac{\partial^n}{\partial x^n}\exp(f(x))=\exp(f(x))\cdot[????]$$
11
votes
1answer
138 views

Differentiation of a function $f:\mathbb{Q}\to \mathbb{Q}$(Rational Calculus)

Assume that $f:\mathbb{Q}\to \mathbb{Q}$ is given such that $\forall a\in \mathbb{Q}$ the following limit, exists \begin{equation} \lim_{x\to a} \frac{f(x)-f(a)}{x-a}\in \mathbb{R} ...
11
votes
3answers
495 views

A limit with an intuitive and wrong answer

In my last question I asked about a limit used in my exploration of tangent circles and whatnot. I decided to come up with a more direct approach to my problem, and now I only have to evaluate the ...
11
votes
3answers
156 views

Applications of functions of the form $f(x)^{g(x)}$

Early on in my calculus education, I learned how to take the derivative of $x^x$ by re-writing it in the form $e^{x\ln x}$. More generally, this technique is helpful in finding the derivative of ...
11
votes
1answer
345 views

Interchanging pointwise limit and derivative of a sequence of C1 functions

Assume the following: $f_n$ is a sequence of $C^1$ functions on $[0,1]$ $f_n(x) \rightarrow 0$ pointwise. $f'_n(x) \rightarrow g(x)$ pointwise. Is it true that $g(x) = 0$ almost everywhere? I think ...
11
votes
1answer
118 views

Derivatives of the Struve functions $H_\nu(x)$, $L_\nu(x)$ and other related functions w.r.t. their index $\nu$

There are some known formulae for derivatives of the Bessel functions $J_\nu(x),\,$$Y_\nu(x),\,$$K_\nu(x),\,$$I_\nu(x)\,$with respect to their index $\nu$ for certain values of $\nu$, e.g. ...
11
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3answers
388 views

Are there “differential equations” involving derivations in the sense of abstract algebra?

There is this abstract notion of a derivation, which really only cares about the property $$D(ab)=aD(b)+D(a)b,$$ where $a,b$ are elements of some algebra. This only tangents the ideas, which lead to ...
11
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1answer
140 views

If $f<1$, $f(0)^2 + f'(0)^2=4$, exists $x_0$ s.t. $f''(x_0) + f(x_0)=0$

Suppose $f:\mathbb{R}\to\mathbb{R}$ is $C^2$, $f < 1$ for all $x$, and $f(0)^2 + f'(0)^2=4$. Show that $\exists x_0$ s.t. $f''(x_0) + f(x_0)=0$. So far, I have let $\phi(x) = f(x)^2 + f'(x)^2$. ...
10
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8answers
3k views

Why do you need to use the chain rule in differentiation of ln?

I understand application of chain rule in the differentiation of a random function $(x^2+3)^3$. But, why do you need to use chain-rule when differentiating something like $\ln(2x-1)$; why won't it ...
10
votes
7answers
1k views

What's wrong with these equations? [duplicate]

My friend Boris (Boryan) gave me a task, and completely refuses to give the answer what's wrong here. $$x^2=\overbrace{x+\cdots+x} ^{x\text{ times}}$$ $$(x^2)'=(x+\cdots+x)'$$ $$2x=1+\cdots+1$$ ...
10
votes
4answers
755 views

Showing $f'(x) = f(x)$ implies an exponential function [duplicate]

Possible Duplicate: Proof that $\exp(x)$ is the only function for which $f(x) = f'(x)$ How can I show the statement $f'(x) = f(x)$ implies the function is defined as $f: \mathbb{R} ...
10
votes
3answers
325 views

Value of $f'(0)$ if $f(x)=\frac{x}{1+\frac{x}{1+\frac{x}{1+\ddots}}}$

Consider the function $$f(x)=\cfrac{x}{1+\cfrac{x}{1+\cfrac{x}{1+\ddots}}} $$ Determine the value of $f'(0)$. I tried to differentiate $f(x)$ but it is not subject to chain rule, and now I'm stuck. ...
10
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4answers
226 views

Computing $\frac{d^k}{dx^k}\left(f(x)^k\right)$ where $k$ is a positive integer

Does anyone know a formula for the derivative $$\frac{d^k}{dx^k}\left(f(x)^k\right)$$ where $k$ is some positive integer? I started trying to work it out but it got messy.
10
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3answers
951 views

Why does the condition of a function being differentiable always require an open domain?

Going through Spivak's Calculus on Manifolds and in his definition of a differentiable function from a subset $A$ of $\mathbb{R}^n$ to $\mathbb{R}^m$, $f$ is said to be differentiable if it can be ...
10
votes
5answers
594 views

Proof for $\sin(x) > x - \frac{x^3}{3!}$

They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried: ...
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 ...
10
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3answers
496 views

$\nabla \cdot \color{green}{(\mathbf{F} {\times} \mathbf{G})} $ with Einstein Summation Notation [Stewart P1068 16.5.27]

$\nabla \cdot \color{green}{(\mathbf{F} {\times} \mathbf{G})} = \partial_h\color{green}{\epsilon_{hij}F_iG_j}$ $ = \epsilon_{hij}\partial_h[F_iG_j]$ $ = \color{purple}{\epsilon_{hij}G_j\partial_hF_i} ...
10
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2answers
125 views

Can you prove My conjecture about Invertiblity of the Derivative Matrix ?! (to use Inverse function Theorem)

In the Analysis2 midterm exam, we had the following problem: Let the equation $a_nx^n+\cdots+a_1x+a_0=0$ has $n$ simple real roots (distinct) $\{\alpha_1,\cdots,\alpha_n\}$. Prove that the above ...
10
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1answer
279 views

Maxima of the function $\left \vert \int_{-1}^1 e^{i(ax+bx^2)}dx \right \vert$

I am looking for extrema of the function $$g(a,b):=\left \vert \int_{-1}^1 e^{i(ax+bx^2)}dx \right \vert$$ where $a,b >0$ are real parameters. I already plotted this function and got the ...
10
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2answers
435 views

geometric interpretation of Lie bracket

On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written: We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to which the integral ...
9
votes
4answers
1k views

Is the determinant differentiable?

I was wondering, given an $n\times n$ square matrix with $n^2$ many entries, the function $\det:\left(a_1,a_2,\ldots,a_{n^2}\right)\to \textbf{R}$ which gives the determinant where $a_{k}$'s are the ...
9
votes
2answers
415 views

Why aren't integration and differentiation inverses of each other?

Integration is supposed to be the inverse of differentiation, but the integral of the derivative is not equal to the derivative of the integral: $$\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\int ...
9
votes
3answers
308 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 ...
9
votes
5answers
661 views

Evaluate the general infinite square root

$$x = \sqrt{n\sqrt{n\sqrt{n}} \cdots}$$ I see that: $$x = \sqrt{nx}$$ $$x^2 -nx = 0$$ Them: $$x(x - n) = 0 \implies x \in \{0, n\}$$ How should I reject the $x = 0$ solution? (any level proof ...
9
votes
6answers
8k 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 ...
9
votes
2answers
1k views

Derive or differentiate?

When the action is: Taking the derivative what verb should be used? to differentiate to derive I feel that deriving is not the correct word here. In my mind it's more a synonym of deducing. Am I ...
9
votes
2answers
471 views

Derivative of $f(x) = (x+x)$

I'm trying to teach myself algebra and derivatives. I learned the derivative for $f(x) = x^2$ from a lesson, and now I thought I would see if I could figure out the derivative of $f(x) = x+x$ on my ...
9
votes
1answer
367 views

Does there exist a continuously differentiable function with the following properties?

Does there exist a continuously differentiable function $f: [1,5] \rightarrow \mathbb{R}$, such that $f(1) \lt 0, f(5) \gt 3$ and $f'(x) \leq e^{-f(x)}$? Now do I just integrate it to get $f(x) ...
9
votes
6answers
128 views

How to differentiate $y=\sqrt{\frac{1+x}{1-x}}$?

I'm trying to solve this problem but I think I'm missing something. Here's what I've done so far: $$g(x) = \frac{1+x}{1-x}$$ $$u = 1+x$$ $$u' = 1$$ $$v = 1-x$$ $$v' = -1$$ $$g'(x) = \frac{(1-x) ...
9
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2answers
231 views

$f'$ exists, but $\lim \frac{f(x)-f(y)}{x-y}$ does not exist

Suppose $f$ is differentiable at $a$, i.e. $\lim_{x\to a}\frac{f(x)-f(a)} {x-a}$ exists. I wondered whether it was necessarily true that $$\lim_{\substack{x,y\to a\\x\neq y}}\frac{f(x)-f(y)}{x-y} ...
9
votes
1answer
132 views

Derivative of a generalized hypergeometric function

Let $$f(a)={_2F_3}\left(\begin{array}c1,\ 1\\\tfrac32,\ 1-a,\ 2+a\end{array}\middle|-\pi^2\right).$$ How to find $f'(0)$ in a closed form?
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votes
2answers
2k views

“Reciprocal” of derivatives

Let $x,y$ be 2 variables. When then is ${dx\over dy }= {1\over {dy\over dx}}$? I guess it is true for total derivatives, but am not entirely sure. What about if the derivatives are only partial ...
9
votes
2answers
133 views

Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ that is differentiable only at $0$ and at $\frac{1}{n}$, $n \in \mathbb{N}$?

How to determine the existence of the function $f: \mathbb{R} \rightarrow \mathbb{R}$, which is differentiable only at $0$ and at $\frac{1}{n}$, $n \in \mathbb{N}$? It's more than enough to give an ...
9
votes
3answers
670 views

Is the mean of the truncated normal distribution monotone in $\mu$?

I am wondering whether the mean of the truncated normal distribution is always increasing in $\mu$. The untruncated distribution of $x$ is $\mathcal{N}(\mu,\sigma^2)$. The mean of the truncated ...
9
votes
1answer
311 views

Infinite Series -: $\psi(s)=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+.+.+ $.

We have a given converging series using derivatives and matrices(Analogue to Taylor's series) $\psi(s)_{3 \times 3}=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+..+.. \tag 1$. ...
9
votes
2answers
313 views

Why are fractal curves nowhere differentiable?

I am a highschool student who stumbled upon fractals when doing a math project. In my research about fractals, I have found that they are nowhere differentiable. Can someone explain this in simple ...
9
votes
2answers
489 views

Evaluating $\int_{0}^{x} e^t \sqrt{2 + \sin(2t)} \, dt$

I was recently asked to evaluate the following integral: $$\int_0^x e^t \sqrt{2 + \sin(2t)} \, dt$$ It was beyond the ken of WolframAlpha, which I find quite discouraging. Does anyone have an idea ...