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

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10
votes
4answers
210 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
236 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
197 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 ...
9
votes
6answers
918 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 $\dfrac{d}{dx}{\large\left(e^{e^x}\right)}\,$?
9
votes
2answers
462 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
3answers
384 views

FoxTrot Bill Amend Problems

So I found this on the Wolfram website today: So I was wondering about how one might be able to (if possible) solve those four problems by hand. Here are the problems, $\LaTeX$ed: $ \lim_{x \to ...
9
votes
3answers
455 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
2answers
163 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
468 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
321 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
votes
1answer
245 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{1-t^2}\left(t^2\right)^{k}\,e^{2\beta t^2},\\ ...
9
votes
2answers
310 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 ...
8
votes
5answers
717 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
876 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
2answers
3k 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 ...
8
votes
3answers
601 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
237 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
273 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 ...
8
votes
3answers
948 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
406 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
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 ...
8
votes
2answers
513 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?
8
votes
5answers
4k 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
3answers
4k 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
247 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) ...
8
votes
3answers
234 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
326 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 ...
8
votes
1answer
57 views

How do I find $f(0)$, $f'(0)$, and $f'(x)$ given $f(x+y)=f(x)+f(y)+x^2y+xy^2$ and $\lim_{x\to0}\frac{f(x)}{x}=1$?

How can I find $f(0)$, $f'(0)$, and $f'(x)$ given that $f(x+y)=f(x)+f(y)+x^2y+xy^2$ and $\lim_{x\to0}\frac{f(x)}{x}=1$.
8
votes
2answers
136 views

Interesting Differentiation Technique

@HansEngler Left the following response to this question regarding "bad math" that works, Here's another classical freshman calculus example: Find $\frac{d}{dx}x^x$. Alice says "this ...
8
votes
3answers
117 views

Find a smooth function $f:\mathbb{R}\to\mathbb{R}$ such that $|f'(x)| < 1$ and $f(x) \neq x$ for all $x\in\mathbb{R}$

Exercise: Find a smooth function $f:\mathbb{R}\to\mathbb{R}$ such that $|f'(x)| < 1$ and $f(x) \neq x$ for all $x\in\mathbb{R}$ I got this exercise from the book "Curso de Análise: volume 1", by ...
8
votes
8answers
544 views

help me understand derivatives and their purpose

I am only starting learning calculus and it's difficult for me to understand the main concept behind calculus ideas particularly differentiation I have searched many resources but most of them are ...
8
votes
2answers
188 views

Prove identity concerning successive derivative of $e^{x^2/2a}$

Prove the following identity: \begin{equation} \left[\frac{d^{2n}}{dx^{2n}}e^{x^2/2a}\right]_{x=0}=\frac{(2n-1)!!}{a^n} \end{equation} The final derivative must be evaluated at $x=0$.
8
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 ...
8
votes
2answers
6k 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 question $w = f(x,y) = x^2 + y^2$ where $x = r\sin\theta $ and $y = r\cos\theta$ requiring the solution to $\partial w ...
8
votes
3answers
71 views

Show that it is possible that the limit $\displaystyle{\lim_{x \rightarrow +\infty} f'(x)} $ does not exist.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ a differentiable function with continuous derivative and the limit $\displaystyle{\lim_{x \rightarrow +\infty} f(x) }$ exists. Show with an example that it ...
8
votes
1answer
153 views

Need a general formula for $\frac{d^n}{dx^n}\left(f(x)^m\right)$

Let $m,n\in\mathbb{N}$. I need to express the derivative $\displaystyle\frac{d^n}{dx^n}\left(f(x)^m\right)$ in terms of sums/products of the derivatives of the function $f$ itself. Here are results ...
8
votes
3answers
194 views
+200

Determining the maximum value for the solution of this delay differential equation?

I am working on the following delay differential equation $$\frac{df}{dt}=f-f^3-\alpha f(t-\delta)\tag{1},$$ where $\frac{1}{2}\leq\alpha\leq 1$ and $\delta\geq 1$. I know that there are three ...
8
votes
1answer
191 views

Does $f\colon \Omega \to \mathbb R$ differentiable imply $f$ locally Lipschitz?

Let $f\colon \Omega \subseteq \mathbb R^n \to \mathbb R$ be a differentiable function. Is it true that $f$ is locally Lipschitz, i.e. Lipschitz on every compact $K \subset \Omega$? If $f$ were ...
8
votes
1answer
99 views

Derivative definition

Let $f$ be a differentiable function in $x_0$. Calculate the following $\lim$: $$\lim_{h\to 0}\frac{f(x_0+2h)-f(x_0-h)}{5h}$$ since we know from theory that $f'(x_0)=\lim_{h\to ...
8
votes
1answer
853 views

Functions whose derivative is the inverse of that function

Everyone knows that there are at least three functions whose derivative is the function itself, namely $e^x, \ 0$ and $-e^{x}$. ( are there more?) I was drawing some polynomials and their ...
8
votes
0answers
81 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. ...
7
votes
2answers
330 views

Is there an analytic function satisfying $\,\,f\big(\!\frac 1 n\!\big)=\frac 1 {\sqrt{n}},\, \,n\in\mathbb N$?

Is there a function that is analytic in an open neighbourhood of $z=0$ and satisfies $$f\left(\!\dfrac 1 n\!\right)=\dfrac 1 {\sqrt{n}},$$ for all natural number $n$? I guess this problem requires ...
7
votes
6answers
382 views

Finding the $n$-th deriviative of $f(x) =e^x \sin x$, solving the recurrence relation

I was given a homework assignment to find a closed solution for the nth deriviative of the function: $f(x) = e^x \sin x$ So far I have been able to obtain the derivative as: $f^{(n)}(x) = e^x S_n ...
7
votes
3answers
261 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. ...
7
votes
2answers
521 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 ...
7
votes
4answers
624 views

If $f(x/2)=f(x)/2$, then $f(x)=f'(0)x$

Let $f:\mathbb R \to \mathbb R$ be differentiable such that $f(x/2)=f(x)/2$ for any $x\in \mathbb R$. How can I prove that $f(x)=f'(0)x$, for any $x\in \mathbb R$? It seems easy, but I don't know why, ...
7
votes
4answers
332 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 ...
7
votes
3answers
178 views

Calculating value of $1000^{th}$ derivative at $0$.

I need to calculate value of $1000^{th}$ derivate of the following function at $0$: $$ f(x) = \frac{x+1}{(x-1)(x-2)} $$ I've done similar problems before (e.g. $f(x)= \dfrac{x}{e^{x}}$) but the ...
7
votes
4answers
346 views

How do I find $\frac{\text{d}}{\text{d}z}\left(z\bar{z}\right)$?

I am seeking $\frac{\text{d}}{\text{d}z}\left(z\bar{z}\right)$ where $f(z)=z\bar{z}.$ And I know that I need to use the following definition of the derivative: $$f'(z)=\lim_{\Delta z\to ...
7
votes
3answers
611 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 ...