Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).
6
votes
3answers
104 views
Is odd continuous function differentiable at $x=0$?
Suppose that $f(x)$ is continuous and odd: $f(-x) = - f(x)$.
Does it have a derivative at $x=0$?
Here is what I got so far: First we calculate $f(0)$ using $f(-0) = -f(0)$, from which $f(0) = 0$.
...
6
votes
2answers
89 views
Calculating with limits
I got this exercise which I quite frankly can't wrap my head around
$$\lim_{x\rightarrow 0} \frac{\ln(1-x)+\sin(x)}{x^2 e^x} $$
The result should be:$\ -\frac{1}{2} $
I tried by derivating the ...
6
votes
3answers
206 views
Is the complex derivative “speed”?
The first thing I was told about the real derivative is that it's "how fast the function is growing" at a given point. This interpretation wasn't addressed in my complex analysis classes. Can the ...
6
votes
4answers
98 views
Math question please Rolle theorem?
I have to prove that the equation $$x^5 +3x- 6$$ can't have more than one real root..so the function is continuous, has a derivative (both in $R$) . In $R$ there must be an interval where $f'(c)=0$, ...
6
votes
3answers
143 views
How is this function never decreasing?!
What I'm doing is finding where this function is decreasing or increasing.
Here is the original function:
$f(x) = \ln(x+6)-2$
I take the prime when I believe is:
$f'(x)= \dfrac{1}{x+6}$
Then I ...
6
votes
2answers
55 views
Trigonometry inflection point
Can anyone help me find the points of inflection in the following function in the interval between $0$ and $2\pi$
$f(x)=\sqrt{2}x^2-4\sin(x)$
for my first derivative I got
...
6
votes
2answers
160 views
Translating intuition into rigor. The chain rule.
When considering two functions $f(x)$ and $g(x)$, it is known that
$$\left(f\circ g(x)\right)' = f'\circ g(x)\cdot g'(x)$$
So my intuitive approach is:
$$\mathop {\lim }\limits_{\Delta ...
6
votes
2answers
291 views
“Reciprocal” of derivatives
Let $x,y$ be 2 variables.
When then is ${dx\over dy }= {1\over {dy\over dx}}$? I guess it is true for total derivatives, but am not entirely sure.
What about if the derivatives are only partial ...
6
votes
1answer
565 views
“Strong” derivative of a monotone function
It is well known that if a function $f\colon \mathbb{R} \to \mathbb{R}$ is monotone then $f'$ exists almost everywhere.
Is it true that if $f$ is monotone then there exists (edit: I mean exists ...
6
votes
1answer
69 views
Derivative definition
Let $f$ be a differentiable function in $x_0$. Calculate the following $\lim$:
$$\lim_{h\to 0}\frac{f(x_0+2h)-f(x_0-h)}{5h}$$
since we know from theory that $f'(x_0)=\lim_{h\to ...
6
votes
1answer
182 views
notation of derivation in differential geometry
I can't wrap my head around notation in differential geometry especially the abundant versions of derivation.
Peter Petersen: Riemannian Geometry defines a lot of notation to be equal but I don't ...
6
votes
1answer
82 views
Minimal definition of the derivative
The definition of the Fréchet derivative according to Wikipedia is:
Let $V$ and $W$ be Banach spaces, and $U\subset V$ be an open subset of $V$. A function $f : U \to W$ is called Fréchet ...
6
votes
2answers
42 views
Free groups and derivative
Per definition a derivative on a group $G$ is a mapping $D:G\rightarrow\mathbb{Z}G$ such that $D(gh)=D(g)+gD(h)$. Now my question:
uppose $G$ is a free group $F=F(X)$ with $X$ a finite set and suppose ...
6
votes
1answer
70 views
Compute the differential of a smooth map
Let $S\subseteq \mathbb{R}^3$ be an oriented regular surface and let $N$ be a field of normal unitary vector on $S$. We consider the map $F:S\times \mathbb{R}\rightarrow \mathbb{R}^3$ defined by ...
6
votes
1answer
103 views
Weaker assumption to ensure $f_{n}^\prime \to f'$
I learned in my real analysis class that if $f_n:[a,b] \to \mathbb{R}$ is a sequence of differentiable functions such that $f_n \to f$ uniformly and $f_{n}^\prime \to g$
uniformly then $f$ is ...
5
votes
6answers
310 views
How do I calculate the derivative of $x|x|$?
I know that $$f(x)=x\cdot|x|$$ have no derivative at $$x=0$$ but how do I calculate it's derivative for the rest of the points?
When I calculate for $$x>0$$ I get that $$f'(x) = 2x $$
but for $$ x ...
5
votes
4answers
246 views
Differentiating $\;y = x a^x$
My attempt:
$$\eqalign{
y &= x{a^x} \cr
\ln y &= \ln x + \ln {a^x} \cr
\ln y &= \ln x + x\ln a \cr
{1 \over y}{{dy} \over {dx}} &= {1 \over x} + \left(x \times {1 \over ...
5
votes
2answers
352 views
Why use the derivative and not the symmetric derivative?
The symmetric derivative is always equal to the regular derivative when it exists, and still isn't defined for jump discontinuities. From what I can tell the only differences are that a symmetric ...
5
votes
3answers
158 views
Find the value of the function at the given point.
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function satisfying the conditions
$$\begin{align*}
(1)&f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}\\
(2)&f(0)=1\\
(3)&f'(0)=-1
...
5
votes
3answers
204 views
How to take the derivative of $x!$
This is the problem I am trying to solve:
$$\lim_{x \to \infty} \frac{e^x x!}{x^x\sqrt{x}}.$$
I believe this is an indefinite form, thus use L'Hospitals's rule.
But the problem I am having is how ...
5
votes
2answers
195 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 ...
5
votes
2answers
205 views
Why isn't $\frac{\mathrm{d} }{\mathrm{d} x} \ln(x)$ specified as $\frac{1}{x},x>0$?
As I understand, $\begin{eqnarray} \frac{\mathrm{d}}{\mathrm{d}x}\ln(x)\end{eqnarray} $ is generally specified as $\begin{eqnarray} \frac{1}{x} \end{eqnarray}$. Would it be more appropriate to state ...
5
votes
2answers
345 views
Implicit Differentiation
I was just wondering where the y'/(dy/dx) in implicit differentiation comes from.
$$
x^2 + y^2 = 25
$$
$$
(d/dx) x^2 + (d/dy) y^2 **(dy/dx)** = 25 (d/dx)
$$
$$
2x + 2y (dy/dx) = 0
$$
$$
(dy/dx) = -x/y
...
5
votes
3answers
261 views
Proof that $Γ'(1) = -γ$?
I know that $Γ'(1) = -γ$, but how does one prove this?
Starting from the basics, we have that:
$$Γ(x) = \int_0^\infty e^{-t} t^{x-1} dt$$
How do we differentiate this? How do we then find that
...
5
votes
3answers
184 views
Understanding if $\sin x = t$ then $\cos x dx = dt$
Hi I need help in understanding this:
if $\sin x = t$ then $\cos x dx = dt$.
My math book mostly uses the Lagrange's notation (prime) and I think I may not fully have grasped the $\frac{dx}{dy}$ way ...
5
votes
3answers
125 views
Uniform Continuity and Differentiation
Is the following true or false?:
Let $f\colon [0,1) \to \mathbb{R}$ be a function differentiable in $[0,1)$ (where the derivative at zero means "right derivative") such that both $f$ and ...
5
votes
4answers
103 views
$k(tx,ty)=tk(x,y)$ then $k(x,y)=Ax+By$
A friend asked me today the following question:
Let $k(x,y)$ be differentiable in all $\mathbb{R}^{2}$ s.t for every
$(x,y)$ and for every $t$ it holds that $$k(tx,ty)=tk(x,y)$$ Prove that
...
5
votes
1answer
208 views
Wrong Wolfram alpha result?
I have this function
$$
f(x,y) = \left\{
\begin{array}{ll}
\frac{x^3}{x^2 + y^2} & \mbox{if } (x,y) \neq (0,0) \\
0 & \mbox{if } (x,y) = (0,0)
\end{array}
\right.
$$
And I want to ...
5
votes
2answers
241 views
Axiomatic approach to differential calculus?
Normally we develop differential calculus by defining a derivative constructively as something related to a quotient (maybe an $\epsilon-\delta$-style limit of a quotient, or maybe the standard part ...
5
votes
2answers
56 views
Inverse and derivative of a function [duplicate]
Find an example of an inverse function f(x) such that its derivative is the same as its inverse.
I tried many different functions but non of them worked.
5
votes
1answer
294 views
Proof for the funky trace derivative : $d (\operatorname{trace} (ABA'C))$?
Given three matrices $A$, $B$ and $C$ such that $ABA^T C$ is a square matrix, the derivative of the trace with respect to $A$ is:
$$
\nabla_A \operatorname{trace}( ABA^{T}C ) = CAB + C^T AB^T
$$
...
5
votes
2answers
141 views
Prove $\lim \limits_{x \to\infty } \frac{f(x)}{x}=0$ and $f$ differentiable implies $ \lim \limits_{x \to\infty } \inf |f'(x)|=0 $
Given a differentiable function on $(a,+\infty)$ such as $\lim \limits_{x \to\infty } \frac{f(x)}{x}=0$ prove the following:
$$ \lim \limits_{x \to\infty } \inf |f'(x)|=0 $$
I just can't see how to ...
5
votes
2answers
128 views
Differentiating under the integral sign problem
Knowing that $$\int_0^\infty e^{-x^2}\,dx = \frac{\sqrt{\pi}}{2},$$
evaluate the integral $$\int_0^\infty e^{-x^2y+1}\,dx.$$
for $y > 0$
5
votes
2answers
213 views
Is there any graphical explanation of the derivative of $\sin x$?
I'm trying to understand in a practical/graphical view the derivative of $\sin(x)$ (that results in $\cos(x)$).
Is there any animation or illustration explaining that?
5
votes
1answer
79 views
Order of growth of derivatives at given x
Is there such an $f$ smooth function and $x\in D_f$, so that the sequence $f(x), f'(x), f''(x), ...$ grows faster than exponential?
Can it grow at a factorial rate or faster?
5
votes
1answer
141 views
Application for mean value theorem
$f(x)$ is three-times differentiable on $[a,b]$, how to show that there is $\varepsilon\in(a,b)$ such that
$$f(b)=f(a)+\cfrac{1}{2}(b-a)[f'(a)+f'(b)]-\cfrac{1}{12}(b-a)^3f'''(\varepsilon)$$
5
votes
1answer
874 views
What exactly is the difference between a derivative and a total derivative?
I am not too grounded in differentiation but today, I was posed with a supposedly question $w = f(x,y) = x^2 + y^2$ where $x = r\sin\theta $ and $y = r\cos\theta$ requiring the solution to $\partial w ...
5
votes
1answer
186 views
Poisson summation formula and Schwartz functions
I am reading a proof of the Poisson summation formula which states that (with my version of the Fourier transform - I think they sometimes vary by a constant factor) for $f$ a Schwartz function on ...
5
votes
1answer
125 views
Higher mixed partial derivatives of $e^{f(x)}$
I have a function $g(x) = e^{-f(x)}$, $x = (x_1,x_2,...,x_n)$. Is there some compact and beautiful formula for the derivative $\frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}...\partial ...
5
votes
2answers
627 views
Finding the Derivative of |x| using the Limit Definition
Please Help me derive the derivative of the absolute value of x using the following limit definition.
$$\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}
$$
I have no idea as to how to ...
5
votes
3answers
44 views
Sum polynomial and derivative
How to prove that if polynomial $W(x)$ has $n$ real roots then
$\forall a \in \mathbb{R}$
$a W(x)+W'(x)$ has more than $n-1$ roots
I have no idea how to solve. Please some hint.
5
votes
1answer
211 views
Show something using the Mean Value Theorem
I've got a exercise about differentiability, mean value theorem and suprema.
To be honest I don't understand the structure of this question. Maybe you guys are so kind to help me out :)
Let $f: ...
5
votes
2answers
129 views
Can you verify my ideas here with the chain rule using limits?
I've been working on understanding limits thoroughly, so I'm rewriting how I understand the chain rule. Please help me fill in my gaps in understanding.
$f$ is some function. Then
$f'(x) = ...
5
votes
1answer
85 views
For a differentiable map $\Phi$ between manifolds $M$ and $W$, what is $d\Phi?$ (Aubin)
I can't understand a passage from A Course in Differential Geometry by T. Aubin. First, there is Definition 2.6., which I posted in this question. And now there's this:
$(\Phi_*)_P$ is nothing ...
5
votes
1answer
134 views
problem on continuous and differentiable function
$f$ is continuous on $[0,\infty)$ and differentiable on $(0,\infty)$ and $f(0) \geq 0$ and $f^\prime(x) \geq f(x)$
to show $f(x)\geq 0 \forall x \in (0,\infty)$
my answer: if $\exists x_0 \in ...
5
votes
1answer
659 views
Derivatives of functions involving absolute value
I noticed that if the absolute value definition $\lvert{x}\rvert=\sqrt{x^2}$ is used then we can get derivatives of functions with absolute value, without having to redefine them as piece-wise.
For ...
5
votes
1answer
104 views
Algebraic transformations to continuously extend functions
Lately I was browsing through my analysis lecture notes (since right know I'm somewhat rusty in analysis) and the proof that $x \mapsto \frac{1}{x}$ is differentiable at every $x'\neq 0$ captured my ...
5
votes
0answers
118 views
Partial derivative of a composite function $\mathbb{R}^n \to \mathbb{R}^n$
I am trying to understand a proof but I am stuck on this technical bit:
Apart from the small typo highlighted, I don't really see how to get the big formula for the partial derivative of $v_i$
...
5
votes
0answers
194 views
Closed form expression for constants
We have the constants $c_{k,n}$ defined by :
$$c_{k,n}=\frac{d^{k}}{ds^{k}}\left(\frac{e^{\frac{1}{n(ns-1)}}e^{\psi\left(\frac{s-1}{s} \right )}}{s} \right )$$
Where $\psi(s)\;$ is the Digamma ...
5
votes
0answers
196 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 ...
