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

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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
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2answers
52 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
11 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
31 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.
3
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1answer
16 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
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1answer
29 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
23 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
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1answer
58 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
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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
51 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 ...
1
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2answers
83 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
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1answer
67 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 ...
0
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1answer
28 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
28 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
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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 ...
2
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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 ...
2
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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
460 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 $x_0\in [0,x]$. ...
1
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0answers
45 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 ...
1
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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 ...
0
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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 ...
2
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2answers
81 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 ...
2
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1answer
28 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
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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
62 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) \ ...
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0answers
31 views

Why aren't integration and differentiation inverses of each other?

Integration is supposed to be the inverse of differentiation, but the integral of the derivative is not equal to the derivative of the integral: $$\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\int ...
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1answer
23 views

second derivative of a parametric equation

can someone please explain how in the proof for the second differential of a parametric function we get from to ? how do we calculate $\frac {d}{dt}$?
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2answers
75 views

What is difference between all of these derivatives?

In calculus II we were introduced to a bunch of new derivatives: the gradient, the derivative $D=\begin{bmatrix} \partial_{x_1} \\ \partial_{x_2} \\ \vdots \\ \partial_{x_n}\end{bmatrix}$, the ...
0
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1answer
41 views

how to prove that a function is not complex differentiable

I was working on a problem on the complex differentiability of the following function: $f(z)= z \operatorname{Re}(z)$. How to find the points where the given function is not differentiable. My ...
1
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2answers
29 views

Finding the minimum point looks easy with a graph but hard with a formula

My research has lead me to the following function: $$ \frac{\sin(x) [\sin^2(x)\cdot F+ \cos^2(x)/F ]} { 1 - \cos(x) } $$ $F$ is a parameter, and I would like to find the minimum value of this ...
1
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1answer
25 views

How to find $\lim\limits_{z \to z_0} \frac{{\overline z}^2-{\overline z_0}^2}{z-z_0}$

Fairly simple question, I want to find this limit $$\lim_{z \to z_0} \frac{{\overline z}^2-{\overline {z_0}}^2}{z-z_0}$$ The original question was to find the region at which the function ...
2
votes
0answers
25 views

Finding the flow of a pushforward of vector field (small edit needed as well)

Let $\mathbb{X}$ be the vector field on $\mathbb{R}^2$ given by $$ \mathbb{X}(x,y) = (y,x). $$ Compute the flow $\Phi_t$ of $\mathbb{X}$. Let $F:\mathbb{R}^2\to \mathbb{R}^2$ be the ...