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

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12 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 $x = 1.3$ ...
0
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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 ...
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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
17 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
86 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
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
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0answers
39 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
60 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
67 views

Using differentiation [closed]

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
189 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
32 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
17 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 ? ...
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1answer
39 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 $\frac {d}{dt}$?
3
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2answers
78 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 ...
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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 ...
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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
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0answers
30 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 ...
1
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1answer
35 views

Fréchet derivatives of $\sum_{n=1}^\infty x_n^2/n^3 -\sum_{n=1}^\infty x_n^4$

I read that the second order Fréchet derivative $F''(0)$ of linear functional $F:\ell_2\to\ell_2$, where $\ell_2$ is the separable real Hilbert space, defined by ...
3
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2answers
112 views

Rolle's Theorem with roots

Let $f : [a, b] \to \mathbb R$ be $n$ times differentiable and have $n+1$ distinct roots (i.e. solutions of $f(x) = 0$) in $[a,b]$. Show that there is an $x \in [a, b]$ such that the $n^{\text{th}}$ ...
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2answers
14 views

Optimization problem - More Hired - Less Produced Each

Brilliantly realizing that there is a demand for chocolate candy in the shape of an integral sign, you start a business to produce such items. Efficiency experts determine that if you employ 30 people ...
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0answers
25 views

Showing that vector field $\mathbb{Z}$ satisfies $\mathbb{Z}\cdot(\nabla \times \mathbb{Z})=0$,connected to the Frobenius Theorem

Suppose $\mathbb{Z}$ is a smooth vector field on $\mathbb{R}^3$ with $\mathbb{Z}^3(x,y,z) \neq 0$. a) Find functions $f$ and $g$ such that the vector fields $\mathbb{X}=(1,0,-f)$ and ...
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0answers
14 views

Another Problem integrating when attempting a solution with the Poincaré Lemma

a) I think that the answer should be $d\nu=10z dx \wedge dy \wedge dz$ b) and c) are easy. d) This is part I am having troubles with. $\begin{align} i_{\hat{\mathbb{X}}_t}\beta &= ...
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2answers
9 views

Differentiating a vector valued function giving a row vector?

If $f:\mathbb R^n \to \mathbb R$, why is $f'(u)$ a $1 \times n$ row vector? (for any $u \in \mathbb R^n$). Many thanks!
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1answer
21 views

Differentiating both sides of an inequality

Let $I = [0, +\infty)$ and let $f : I \to \mathbb{R}$ be a differentiable function on $I$ such that there exists a constant $M \in \mathbb{R}$ satisfying $$f'(x) \le M,\ \forall x \in I$$ Prove ...
8
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2answers
90 views

does derivative of convergent function go to 0? [duplicate]

My question: if $f$ is differentiable and $\lim_{x \to \infty} f(x) = M$, does this imply that $\lim_{x \to \infty} f'(x) = 0$? My thinking: there exists $X$ such that $\forall x,y>X$, ...
3
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1answer
25 views

Curvature of plane curve, formula disagrees with Mathematica?

I have the following equation $$y=\pi\ln(2x)$$ And when I ask Mathematica/WolframAlpha for the curvature I get $$K=\frac{\pi x}{(x^2+\pi^2)^{3/2}}$$ However the formula for curvature of a plane ...
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2answers
72 views

Use the Mean Value Theorem to show that if $|f'(x)| ≤ C<1$, then $f$ has at most one fixed point

Use the Mean Value Theorem to show that: if $|f'(x)| ≤ C < 1$ $\forall x$, then $f(x) = x$ has at most one solution. So using the Mean Value Theorem I know that $$-1<-C\leq ...
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1answer
45 views

Derivative of scalar function with respect to vector

Suppose I have three constant symmetric matrix $\mathbf{M}_{n\times n}$, $\mathbf{C}_{n\times n}$ and $\mathbf{D}_{n\times n}$ and two variable vectors $\mathbf{q}_{n\times 1}$ and ...
2
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1answer
53 views

Question about length of curve?

The question: Find length of curve defined by $\displaystyle y=2\ln\left[\left(\frac{x}{2}\right)^2-1\right] $ from $x=4$ to $x=6$ Here is the work I have done, but I seem to keep getting it ...
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0answers
27 views

Using Frobenius' Theorem for 3 functions in 2 variables [closed]

i) 1) $v= \frac{\partial u}{\partial x}$ 2) $w= \frac{\partial u}{\partial t}$ 3) $\frac{\partial v}{\partial t}= \frac{\partial w}{\partial x}$ 4) $\frac{\partial v}{\partial x}= \frac{\partial ...
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1answer
23 views

Showing that $F^*(L_{\mathbb{Y}}\omega)=L_{\mathbb{X}}(F^*\omega)$

Let $F:U \rightarrow V$ be a diffeomorphism between open sets in $\mathbb{R}^n$. Let $\mathbb{Y}$ be a vector field on $V$ and $\omega$ a $k$-form on $V$. Show that ...
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0answers
16 views

Borel sets and absolutely continuous functions - second part

Borel sets and absolutely continuous functions This question is a part of the question of this link. So, in order to show that $F'=0$ on set of positive measure, what I did was mentioned here: Let ...
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2answers
33 views

What's the rule for knowns and unknowns when dealing with derivatives

So a rule of thumb when doing basic algebra is you must have as many equations as you have unknowns. For example: $0=4x+6y^2$ $3x=2\sqrt{y}$ You have two equations and two unknowns and thus can ...
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2answers
65 views

For a $C^1$ function, the difference $|{g'(c)} - {{g(d)-g(c)} \over {d-c}} |$ is small when $|d-c|$ is small

Suppose $g\in C^1 [a,b]$. Prove that for all $\epsilon > 0$, there is $\delta > 0$ such that $|{g'(c)} - {{g(d)-g(c)} \over {d-c}} |{< \epsilon }$ for all points $c,d \in [a,b]$ with $0 ...
2
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0answers
45 views

Directional derivative (Vector)

Given $f:\mathbb{R}^2 \to \mathbb{R}^2$ is a map $f(x,y)=(u(x,y),v(x,y))$ and $\alpha=(\alpha_1,\alpha_2)$ is a point, then how does one show that $f$ is differentiable (or not) in the direction ...
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4answers
50 views

Applying math knowledge [closed]

Currently I'm in the middle of my first year of college studying informatics engineering. I was never great at math, but if I put some effort, I understand it and constantly get good grades. However, ...
1
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1answer
27 views

Show that $i_Yi_Xd\omega=d\omega(X,Y)$ for $\omega$ a $1$-form

If $\omega$ is a $1$-form, how does $i_Yi_Xd\omega=d\omega(X,Y)$? I get that $d\omega$ is a 2-form. So $i_X(d\omega)=d\omega(X,v_{2})$. So how do we proceed? I dont see how the step ...
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2answers
45 views

Differentiation under the integral sign, please help

Find the partial derivatives of the function: $$\int_{x^2e^{5y}}^{\ln(x^3-2)}\cos(t^2)dt$$ Maple responds: $$-2\,\cos \left( {x}^{4} \left( {{\rm e}^{5\,y}} \right) ^{2} \right) x {{\rm ...
1
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3answers
53 views

Economically computing $d\beta$

$\displaystyle \beta = z\frac{x dy \wedge dz + y dz \wedge dx + z dx \wedge dy}{(x^2+y^2+z^2)^{2}}$ Show that $d\beta=0$. So, let $r=x^2+y^2+z^2$, $\begin{align} \displaystyle d\beta &= ...
0
votes
1answer
35 views

Two statements about one-sided derivative and monotony

The statement 1 is: $f\colon [a,b]\to\mathbb R$,continuous on $[a,b]$,$f'_-(x)$ exists and is $\le0$ for all $(a,b]$.Can we infer that f is non-increasing on $[a,b]$? My attempt is: Assume $f$ is not ...
2
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4answers
64 views

Taking the derivative of $(1+x^2)^{(\sqrt{x})}$

As stated above, I'm having trouble taking the derivative of $(1+x^2)^{(\sqrt{x})}$. I know that I should somehow be using the exponential derivative form of $\dfrac{d}{dx} ( a^x ) = a^x\ln(a)$, but I ...
0
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2answers
36 views

differentiation of $g(x) = \lvert f(x)\rvert$ where $f(x)$ and $D(f(x)) = 0$

I'm really stumped on this problem and don't know how to go about it. It says $g(x)$ = $|f(x)|$ and to show that if $f(c) = 0$ and g is differentiable at c, then one must have $D(f)(c) = 0$. ...
1
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2answers
40 views

A limit question involving power of positive numbers

I'm trying compute the following limit: $$\lim_{t\to0}\left(\frac{1}{t+1}\cdot\frac{b^{t+1}-a^{t+1}}{b-a}\right)^{1/t},\quad b>a>0.$$ I know $\displaystyle\lim_{t\to0}(1+t)^{1/t}=e$ and ...