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

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6
votes
2answers
63 views

Tough Differential Equation

Solve $$(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$$ I tried the substitution $y^2=t$ ; $2y\:\text{dy}=\text{dt}$ to get $$(x^3)\:\text{dt}+(1-t)[(x^2+1)t-1]\:\text{dx}=0$$ ...
8
votes
0answers
63 views

Function's analytic continuation is its own derivative

This is the question we were asked at the university by our professor for complex analysis. Not as an exam, but as a challenge. I don't think he knew the answer himself. Find a nontrivial example of ...
-2
votes
0answers
50 views

What is the “actual definition” of the following?

Imagine you were standing on the ground and as the ground starts moving you stay exactly where you are. Your change in movement, by standing on a moving surface, is like a path-line. Suppose we take ...
0
votes
4answers
71 views

Differentiating $ \left( 1 - \frac {1}{x} \right)^x $

I have a calculus question. How does one differentiate $\left(1-\frac{1}{x}\right)^x$, for x>1? It should be positive right?
0
votes
2answers
46 views

Why is this map not surjective at the origin?

$f:\mathbb{R}^{n+1} \rightarrow \mathbb{R}$ $f(x) = |x|^2$ Then the derivative map is $df_{x}(v)=2\sum_{i}{x^iv^i}$ is surjective except at 0. Is it because at 0 df only goes to 0, and doesn't ...
-1
votes
1answer
30 views

Finding a differentiable inverse of $f(x)=x/\cos x$

Let $$ f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right) \rightarrow \mathbb{R} $$ be defined by $$ f(x) = \frac{x}{\cos x}. $$ We're supposed to show that $f$ has a differentiable inverse $$f^{(-1)}$$ ...
1
vote
0answers
22 views

Derivative of a composition of function - nice proof

Let's consider the well known "fake" proof below for the derivative of the composition of functions: Let $E,G$ be intervals of $\mathbb{R}$, let $F$ a subset of a normed vector space, let ...
3
votes
1answer
65 views

Derivability of a function with an infinity of zeroes

Let $F$ be a normed vector space and $a\in F$. Is there a non zero function $f:\mathbb{R}\rightarrow F$, such that $f'(a)=0$ and $f$ is $0$ an infinity of times in any neighborhood of $a$ ? If not, ...
1
vote
0answers
29 views

Differentiation of $\exp(A)$

Let's say we have $${\sigma(\exp(a\cdot X^{-1} \cdot a^\mathrm{T}))}/{\sigma X}$$ when I know that the term inside the exponent is essentially a scalar. Should I differentiate according to ...
1
vote
0answers
19 views

derivative of t distribution cdf wrt degrees of freedom

Given the cdf of a t distribution as follows: $T_\nu(x)=\frac{1}{2} + x\Gamma(\frac{\nu+1}{2}) + \frac{_2F_1 ...
3
votes
0answers
31 views

An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
0
votes
3answers
39 views

How to differentiate the function $f(x) = [ \frac{a+x}{b+x}]^{a+b+2x}$?

It has been given that, $$f(x) = \Big[ \frac{a+x}{b+x}\Big]^{a+b+2x}$$ How to prove , $$f'(0) = 2\ln \frac{a}{b}+ \frac{b^2-a^2}{ab}\Big[\frac{a}{b}\Big]^{a+b}$$ Do I have to take the logarithm of ...
0
votes
1answer
39 views

If $f$ s twice differentiable and satisfies the following constraints, prove $f'(0)>-\sqrt 2$

Let $f$ be a twice differentiable function on the open interval $(-1,1) $such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0 $and $f''(x) \le f(x)$, for all $ x\ge 0$. Show that ...
0
votes
2answers
62 views

Why doesn't $\ln (x)$ have an asymptote since its derivative is $1/x$?

My understanding is that the derivative gives the gradient of the function at that point. So for the function $x^2$, its gradient at point $x=10$ is equal to $20$. Extrapolating this to $\ln (x)$, ...
1
vote
2answers
32 views

Give an example of a function who is nondifferentiable on (0, 2) but has an antiderivative on (0, 2)

Originally when I was playing around with this problem, I tried to first find a function who was differentiable, but whose derivative was not differentiable at a specific point. So I figured out the ...
0
votes
2answers
58 views

How to find the derivative of $F(x)=\int_{x^2}^{4x^2} \sin \sqrt t\;\;dt$?

For a real number $t>0$, let $\sqrt t$ denote the positive square root of t. For a real number $x>0$, let $F(x)=\int_{x^2}^{4x^2} \sin \sqrt t\;\;dt$. If $F'$ is the derivative of $F$, then ...
0
votes
0answers
16 views

Gateaux variation (Functional Derivative) of functional with convolution

Given the following functional $F[f]=\int f(x) \log(g(x)) dx$ find Gateaux variation. Also, $g(x)$ is given by convolution $g(x)=y(x) * f(x)=\int y(\tau) f(x-\tau) d\tau$, so \begin{align*} ...
1
vote
2answers
34 views

Matrix representation of the derivative of a smooth function

Let $V:\mathbb R^n\to\mathbb R$ be a smooth function and define the Hamiltonian function $H:\mathbb R^n\times\mathbb R^n\to\mathbb R$ (kinetic plus potential energy) by $$H(x,y):=\frac ...
0
votes
1answer
29 views

Express the limit in terms of $f'(x_{0})$

Find the following limit in terms of $f'(x_{0})$: $$ \lim_{h \to 0} \frac{f(x_{0} - 3h) - f(x_{0})} {h} $$ Any help would be appreciated.
3
votes
1answer
19 views

Chain rule notation for composite functions

Suppose I have a function $ f(x, y, g(x, y)) $ How would I express $ \frac{\partial f}{\partial x} $? Using the chain rule, you'd naturally come up with $ \frac{\partial f}{\partial x} + ...
2
votes
1answer
32 views

Derivatives defined on a discrete state space

Ive been looking at certain economic papers, and optimal control papers. They define a state variable, $\omega$, which follows a discrete time Markov Chain. Then they define a utility function ...
0
votes
3answers
24 views

Derivative with Logarithm Problem

I'm not sure how to approach this problem and solve it. $$y=\log_5\ln(x^3+6)^4$$
5
votes
1answer
59 views

If $f(0)=0$ and $f(1)=1$, prove that $\int_0^1 \left |f'(x)-f(x) \right |dx\geq e^{-1}$

Let $f$ be a differentiable function on $[0,1]$ such that $f(0)=0$ and $f(1)=1$. If $f'$ is continuous, prove that $$\int_0^1 \left |f'(x)-f(x) \right |dx\geq e^{-1}$$ Progress I let ...
1
vote
1answer
42 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 ...
2
votes
2answers
53 views

derivative of a symmetric bilinear form (quadratic form version)

Let $A=A^T\in \mathbb R^{k\times k}$ be a nonzero symmetric matrix and define $F:\mathbb R^k\to\mathbb R$ by $$f(x):=x^TAx$$ Then why $df(x)\xi=2x^TA\xi$ for $x,\xi\in\mathbb R^k$?
0
votes
1answer
17 views

uniform convergence of diff'ble functions with derivatives converging in $L^1$

Suppose we have a sequence of differentiable functions $f_n$ from a closed interval $I\subseteq\mathbb{R}$ to $\mathbb{R}$ with the following properties: $f_n$ converges uniformly to some function ...
1
vote
2answers
84 views

Proving that $\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \lvert f(z) \rvert^2 = 4 \lvert f'(z) \rvert^2$

Given $f$ entire show that $$ \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \lvert f(z) \rvert^2 = 4 \lvert f'(z) \rvert^2 $$ I've come close to getting the exact ...
3
votes
1answer
70 views

If $f(0) = f(1)=0$ and $|f'' | \leq 1$ on $[0,1]$, then $|f'(1/2)|\le 1/4$

Let $f : [0,1] \rightarrow \mathbb{R}$ be a function whose second order derivative $f''(x)$ is continuous on $[0,1]$. Suppose that $f(0) = f(1)=0$ and that $|f''(x)| \leq 1$ for any $x \in [0,1]$. ...
0
votes
0answers
12 views

Proving $\frac{d}{d\theta}\mathbb E\left[ \log\left( \frac{AY+BY+N}{ AY+BY \frac{X}{\theta^{-\alpha}} +N } \right) \right] \leq 0$

Let $X$ and $Y$ be exponentially distributed random variables with means $\theta^{-\alpha}$ and $(1-\theta)^{-\alpha}$, respectively. Simulation results suggest that $$\frac{d}{d\theta}\mathbb ...
0
votes
1answer
30 views

Holomorphic function and nth derivative.

Let $K$ be a open connected subset of complex numbers and $f$ holomorphic on $K$. If $f=0$ on some open disc $D$ in $K$, then is it true that $n$th derivative of $f$ is $0$ for all points in $D$ ...
0
votes
0answers
30 views

Chain rule (derivative) for for complex data

I found some difficulties in extending the chain rule for complex data. Any suggestion will be appreciated, thanks. In the complex domain, for example, we have a function ...
1
vote
0answers
29 views

Slope Formula Approaches Value of Derivative at a Point

I came across this question while helping a friend study for an Analysis exam; Analysis is not exactly my forte, so maybe I'm missing something obvious, I don't know: Suppose ...
2
votes
3answers
49 views

How “sharp” does a cusp have to be in order for the equation to be nondifferentiable?

From a mathematical standpoint, I understand the concept of cusps: for example, a cusp exists at the origin of $y=|x|$ because one cannot take the limit from both sides, and therefore the derivative ...
2
votes
3answers
44 views

Graphing $\frac{x^2-x+1}{2(x-1)}$

I need to graph $$\frac{x^2-x+1}{2(x-1)}$$ So I reduced it to make the derivative easy: $$f(x) = \frac{x(x-1)+1}{2(x-1)} = \frac{x}{2} + \frac{1}{2(x-1)}\\f'(x) = \frac{1}{2} - ...
10
votes
3answers
464 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 ...
0
votes
2answers
42 views

Complex Analysis using derivatives

I have been studying Euler's Formula and its derivation. In an article I am reading, I came across a use of derivatives I did not understand and am hoping someone can explain it. The use of ...
1
vote
2answers
41 views

Proof $|\sin(x) - x| \le \frac{1}{3.2}|x|^3$

So, by Taylor polynomial centered at $0$ we have: $$\sin(x) = x-\frac{x^3}{3!}+\sin^4(x_o)\frac{x^4}{4!}$$ Where $\sin^4(x_0) = \sin(x_o)$ is the fourth derivative of sine in a point $x_0\in [0,x]$. ...
1
vote
0answers
46 views

differential $df$ of a function $f(x)$

''Explain what is the differential $df$ of a function $f(x)$ and what is the differential of a function $f(x)$ at a point $x=a$. give proper examples...'' This is just part of the instruction ...
1
vote
0answers
40 views

Newton method for maps between Banach spaces

I am trying to understand the following theorem, which can be found in Kolmogorov and Fomin's (p. 509 here): Let map $F$ [$:X\to Y$ where $X,Y$ are Banach spaces] be strongly differentiable in a ...
0
votes
1answer
16 views

Bounding taylor error

I calculated the polynomial or order $2$ for $\ln(x)$, centered at $x_o=1$, which is: $$\ln(1.3) = \ln(1.0) + \ln'(1.0)(x-1) + \ln''(1.0)(x-1)^2$$ Where the lagrangian error is: $$E(x) = ...
0
votes
0answers
11 views

Functional derivative of $\int f_{X,Y}(x,y)e^{-f_{Y|X}(y|x)} dx$ with respect to $f_X(x)$

What is functional derivative of \begin{align*} \int f_{X,Y}(x,y)e^{-f_{Y|X}(y|x)} dx \end{align*} with respect to $f_X(x)$. Here $f_{X,Y}(x,y)$ is joint probability density of r.v. $(Y,X)$ and ...
1
vote
2answers
51 views

Taylor approximation for $\ln(1.3)$

I have to calculate an approximation for $\ln(1.3)$ using degree $2$ expansion for Taylor polynomial: $$P_2(x) = f(x_0) + f'(x_0)(x-x_0) + f''(x_0)(x-x_0)^2$$ So I can take $x_0 = 1$ and $x = 1.3$ ...
0
votes
0answers
60 views

Does there exist unique $c \in (0,1)$ such that $f'(c)=f(c)$? [duplicate]

If $f: [0,1] \to \mathbb R$ be a continuous function differentiable in $(0,1)$ such that $f(0)=f(1)=0$ then by Rolle's thorem for $e^{-x}f(x)$ , it is evident that $f'(x)=f(x) $ has a solution in ...
0
votes
1answer
22 views

You must cut a wire of $36cm$ to form a triangle and a rectangle in a specific place to find the minimum area.

Well, I have an wire of $36cm$ and I need to cut it in two parts, one to form an equilateral triangle, and the other to form a rectangle such that its width is two times the height. Where do I need to ...
0
votes
0answers
16 views

representation of Eulers's equation in biharmonic form

As we know the Euler's equation $${\rm div}{\rm div}(\frac{\nabla^2F}{\|\nabla^2F\|})=0$$ Can be written in biharmonic equation form $$\Delta^2F+ (something)=0$$ I want to know in the context of solid ...
2
votes
2answers
83 views

Is the function $y = a^x + b$ exponential?

What exactly is an exponential function? Some of the sources at which I looked said that it's a function where the rate of change at $x$ $(f'(x))$ is proportional to the value at that point $(f(x))$, ...
1
vote
2answers
60 views

Showing that $f(x)=x^2$ for $x \in \mathbb{Q}$ and $f(x)=0$ for $x \not\in \mathbb{Q}$ is differentiable in $x=0$

I am supposed to show that $f(x) = x^2$ for $x$ in the rationals and $f(x) = 0$ for $x$ in the irrationals is differentiable at $x = 0$ and I am supposed to find the derivative of $f(x)$ at $x = 0$. ...
1
vote
1answer
31 views

graphing $\frac{x^3-x+1}{x^2}$

I want to graph: $$f(x) = \frac{x^3-x+1}{x^2}$$ so I took the first derivative: $$f'(x) = \frac{x^3+x-2}{x^3}$$ but this function is hard to find the signals. In other words, it's hard to find ...
2
votes
1answer
30 views

Partial Derivatives versus Proper Derivatives

I'm having some difficulty understanding exactly what a partial derivative is. I had been content with the definition $$\frac{\partial F}{\partial x_i } = \lim_{\Delta x \rightarrow 0} \frac{F(x_0, ...
2
votes
0answers
36 views

Calculate the distance between intersection points of tangents to a parabola

Question Tangent lines $T_1$ and $T_2$ are drawn at two points $P_1$ and $P_2$ on the parabola $y=x^2$ and they intersect at a point $P$. Another tangent line $T$ is drawn at a point between $P_1$ ...