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

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1answer
18 views

Derivability of a function with an infinity of zeroes

Let $F$ be a vector space and $a\in F$. Is there a 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, how can one ...
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0answers
28 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 ...
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0answers
17 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 ...
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0answers
21 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 ...
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3answers
36 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 ...
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1answer
30 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 ...
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2answers
53 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 $\ln (x)$, ...
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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 ...
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2answers
56 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 ...
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0answers
15 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*} ...
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2answers
33 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 ...
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1answer
28 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.
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1answer
17 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} + ...
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1answer
30 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 ...
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3answers
24 views

Derivative with Logarithm Problem

I'm not sure how to approach this problem and solve it. $$y=\log_5\ln(x^3+6)^4$$
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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 ...
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1answer
40 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
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2answers
52 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 $x,\xi\in\mathbb R^k$?
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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 ...
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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 ...
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1answer
68 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 $x \in [0,1]$. ...
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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 ...
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1answer
29 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 $D$ ...
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0answers
29 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 ...
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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 ...
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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 ...
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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} - ...
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3answers
462 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 ...
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2answers
41 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 ...
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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 $x_0\in [0,x]$. ...
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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 ...
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0answers
39 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 ...
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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) = ...
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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 ...
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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 $x = 1.3$ ...
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0answers
59 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 ...
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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 ...
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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 ...
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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 $(f(x))$, ...
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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
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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 ...
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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, ...
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0answers
29 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 $P_1$ ...
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1answer
57 views

Show that $f$ is a linear map if $f$ is differentiable and its derivative is constant:

Show that if $f:ℝ^m→ℝ^n$ is a differentiable function whose derivative function $f′$ is a constant function and such that $f(0)=0$, then $f$ is a is a linear map. I am a little lost on this. I know ...
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4answers
63 views

Using differentiation [on hold]

The curve shown below has its equation: $y=3x^5-5x^3$ Find algebraically the coordinates of the points $A$ and $B$. ($7$ mark question)
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1answer
188 views

If $dx/dy =\sin(x)$ then is $dy/dx = 1/\sin(x)$?

If $\dfrac{dx}{dy} = \sin(x),$ then is $\dfrac{dy}{dx} = \dfrac{1}{\sin(x)}$? I'm trying to understand how to manipulate $dx$ and $dy$ quantities effectively.
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0answers
30 views

How can I differentiate this equation? $\sin(x)^{\cos{y}}+\sin{y}^{\cos{x}}=3$

$$\sin(x)^{\cos{y}}+\sin{y}^{\cos{x}}=3$$ I solved a similar equation where those 2 functions were equal to each other by taking the natural log for both sides but now I don't know what to do, taking ...
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0answers
16 views

Diffferentiability of complex functions

I need you help me please. I don't know how solve this Find $f_{z}$ y $f_{\bar{z}}$ where $f(z)=\left |{z}\right |^{2} +\displaystyle\frac{z}{\bar{z}}$ moreover what points is differentiable f ? ...
0
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1answer
37 views

Critical points of $f(x, y, z) = \frac{x^5 + y^5+z^5}{x^2+y^2+z^2}$?

What are the critical points of $f(x, y, z) = \frac{x^5 + y^5+z^5}{x^2+y^2+z^2}$? I get a complicated system of equations which is not linear that I do not know how to solve when I equal the gradient ...
0
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1answer
34 views

Calculus proof attempt

Why does $$\frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{\dot{y}}{\dot{x}}\right)\frac{1}{\dot{x}}$$ not yield the same result as $$ \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{\dot{x}}\right) \ ...