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

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1answer
114 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
1answer
480 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? ...
11
votes
3answers
9k views

What exactly is the difference between a derivative and a total derivative?

I am not too grounded in differentiation but today, I was posed with a supposedly easy question $w = f(x,y) = x^2 + y^2$ where $x = r\sin\theta $ and $y = r\cos\theta$ requiring the solution to ...
11
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3answers
154 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
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1answer
297 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
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3answers
362 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
138 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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6answers
941 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 $\dfrac{d}{dx}{\large\left(e^{e^x}\right)}\,$?
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8answers
2k 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 ...
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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
662 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
4answers
218 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
votes
5answers
310 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
233 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
472 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 ...
10
votes
1answer
272 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
votes
2answers
360 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
2answers
4k views

Differentiating an Inner Product

If $(V, \langle \cdot, \cdot \rangle)$ is a finite-dimensional inner product space and $f,g : \mathbb{R} \longrightarrow V$ are differentiable functions, a straightforward calculation with components ...
9
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3answers
303 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. ...
9
votes
2answers
469 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
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1answer
320 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
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2answers
438 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 ...
9
votes
2answers
215 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
2answers
1k 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
3answers
569 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
304 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
237 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
477 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 ...
9
votes
1answer
394 views

Doubt regarding calculus, graph of functions, point of inflection.

We're studying the application of derivatives in mathematics right now. This refers to a question which arose in my head while solving a particular problem. The problem was: A function $f(x)$ is ...
9
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1answer
264 views

Find compressed form for cumbersome calculation

Given the three functions $u^{\mathrm{(I)}}(t)\;=t \left(t^2\right)^{k}\,e^{2\beta t^2},\\ u^{\mathrm{(II)}}(t)=\sqrt{\left(t^2\right)^{2k}-\left(t^2\right)^{2k+1}}\,e^{2\beta t^2},\\ ...
9
votes
0answers
76 views

What's the minimal structure needed to define a notion of derivative?

I know that, for example, to define a limit all you need is the notion of "closeness" generated by a topology; and to define an integral you need a measure function and a sigma-algebra on which it is ...
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0answers
86 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. ...
8
votes
5answers
751 views

Calculating the shortest possible distance between points

Question: Given the points $A(3,3)$, $B(0,1)$ and $C(x,0)$ where $0 < x < 3$, $AC$ is the distance between $A$ and $C$ and $BC$ is the distance between $B$ and $C$. What is x for the distance ...
8
votes
4answers
988 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 ...
8
votes
3answers
699 views

What does the derivative of area with respect to length signify?

Suppose that we have a square sheet of edge length $L$. Its area $A=L^2$. Differentiating $A$ w.r.t. L, we get $$\dfrac{dA}{dL}=2L$$ I do understand what it means to differentiate, ...
8
votes
3answers
246 views

Is $d^2y/dx$ a valid mathematical notation?

I have often seen "the second derivative of y with respect to x" written as $${d^2y\over dx^2},$$ but I don't understand the reason for this notation. I have always seen it written as $${d^2y\over ...
8
votes
4answers
280 views

Is it known or where does this lead to?

I am eleventh class student, recently I started learning calculus. I was experimenting on various things, and found a new thing. It is as follows. Let us consider a function $f(x)$which is ...
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6answers
6k 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 ...
8
votes
2answers
323 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 $x \in [0,1]$. ...
8
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3answers
1k views

n-th derivative of $\frac{\ln x}{x}$.

Let $f(x)=\frac{\ln x}{x},x>0$. Show that $$f^{(n)}(1)=(-1)^{n+1}(n!)(1+\frac{1}{2}+\cdot+\frac{1}{n})$$ Trial: n-th derivative of $\ln x$ is $$(-1)^{n-1}(n-1)! x^{-n}$$ and n-th derivative of ...
8
votes
2answers
482 views

Is there a differential limit?

I'm wondering if there's such a concept as a "differential limit". Let me give an example because my nomenclature is my own and unofficial, but hopefully indicative of the concept. For some function ...
8
votes
2answers
549 views

Sobolev meets Wiener

Even though the Wiener process (Brownian motion) is continuous, it has no derivative at any point. Does it at least have weak derivatives?
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2answers
764 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 ...
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2answers
712 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 ...
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2answers
90 views

does derivative of convergent function go to 0? [duplicate]

My question: if $f$ is differentiable and $\lim_{x \to \infty} f(x) = M$, does this imply that $\lim_{x \to \infty} f'(x) = 0$? My thinking: there exists $X$ such that $\forall x,y>X$, ...
8
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2answers
501 views

Definite integral $\int_0^{\frac{\pi}{2}}\! \frac{\sin x }{\sqrt{\sin 2x}}\,\mathrm{d} x$

$$\int_0^{\frac{\pi}{2}}\! \frac{\sin x }{\sqrt{\sin 2x}}\,\mathrm{d} x$$ I'm pretty sure I can finish it after finding the anti derivative. I tried changing the denominator to $2\sin x \cos x$ and ...
8
votes
3answers
5k views

Derivative of determinant of a matrix

Good morning everyone, I would like to know how to calculate: $\frac{d}{dt}\det \big(A_1(t), A_2(t), \ldots, A_n (t) \big)$ Help me please. Thank you
8
votes
1answer
543 views

Where is the error in my proof that all derivatives are continuous?

I know that this can not be true due to counter-examples but I don't know where the error in my reasoning is. Assumption: If $f(x)$ is differentiable in $\mathbb{R}$ then the derivative $f'(x)$ is ...
8
votes
3answers
238 views

is the following a decreasing function?

I am stuck on figuring out why the following function is a decreasing function when I read a paper. The function is following $$f(x)=-\frac{1}{x}\log[{pe^{-ax}+(1-p)e^{-bx}}]$$ where $a$ and $b$ are ...
8
votes
3answers
415 views

Given that $x = 4\sin \left( {2y + 6} \right)$ find dy/dx in terms of x

My attempt: $\eqalign{ & x = 4\sin \left( {2y + 6} \right) \cr & {{dx} \over {dy}} = \left( 2 \right)\left( 4 \right)\cos \left( {2y + 6} \right) \cr & {{dx} \over {dy}} = 8\cos ...