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

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2answers
48 views

Linear Algebra Change of Basis problem

So, $\mathbb{P}_2$ is the vector space of all polynomials with degree less than or equal to 2 and that $E=\{1,t,t^2\}$ is a basis for $\mathbb{P}_2$ We define $p_1(t)=1+2t$ $p_2(t)=t-t^2$ ...
-1
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1answer
69 views

How can I prove $x^3\, \frac{d^3 y}{dx^3} = \Delta(\Delta-1)(\Delta-2)y$?

This equation is used to solve Cauchy Euler Equation As it can be seen author has provided explanation of the fact how ...
4
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1answer
107 views

Find a formula for $f''$ in terms of $f$, where $f\gt 0$ and $(f')^2=f-\frac{1}{f^2}.$

Problem: Suppose that a function $f \gt 0$ has the property $$ (f')^2=f-\frac{1}{f^2} $$ Find a formula for $f''$ in terms of $f$. Hint: Use Theorem 7. Theorem 7: Suppose that $f$ is ...
1
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1answer
26 views

Computing $\bigtriangledown r^m$ knowing the vector $\boldsymbol r$

I am asked to compute $\bigtriangledown \cdot \boldsymbol r$ and $\bigtriangledown r^m$ for $m$ constant, where $\boldsymbol r =x \boldsymbol i+ y\boldsymbol j +z\boldsymbol k$ and $r= |\boldsymbol ...
2
votes
1answer
76 views

Prove there exists a unique local inverse.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
0
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1answer
101 views

Prove the following function is Lipschitz with constant less than 1.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
-4
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1answer
107 views

Find partial derivatives, given directional derivatives. [closed]

You are given that the directional derivatives of a function $f$, at the point $(a, b)$, in the direction of the two vectors $(1, 2)$ and $(−1, 1)$, are $2$ and $3$ respectively. Find the partial ...
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1answer
34 views

Varying definitions for concavity of a function

I've been working on concavity of functions and have noticed that different texts define this notion in different ways. Specifically, some texts include the endpoints of an interval when describing ...
1
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2answers
50 views

Polynomial must be monotone between its extrema

Suppose that the polynomial function $f(x)=x^n+a_{n-1}x^{n-1}+\cdots +a_0$ has $k_1$ local maximum points and $k_2$ local minimum points. Show that $k_2=k_1+1$ if $n$ is even, and $k_2=k_1$ if $n$ is ...
1
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2answers
60 views

Maximizing area under $y=e^{−{∣x∣}}$

The coordinates of the point $M(x,y)$ on $y=e^{−{∣x∣}}$ so that the area formed by the coordinates axes and the tangent at $M$ is greatest is what? I tried to plot the graph but after that I'm not ...
1
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0answers
83 views

Calculation of a Frechet derivative

Say I have an infinite sequence $X=(x_i)$, $i=1,2,3,\ldots$ such that it's in $\ell^2$ space, i.e. $\sum_{i=1}^\infty|x_i|^2<\infty$. Now, this function that takes this infinite sequence to a real ...
0
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0answers
52 views

Why doesn't $\dot{x}=-\dot{z}sin(φt)-φzcos(φt)+\dot{y}cos (φt)-φy sin(φt)$

\begin{align} \ {x}=y cos (φt) -zsin(φt) \end{align} \begin{align} \dot{x}=\dot{y}cos (φt)+(-φy sin(φt))-(\dot {z}sin(φt)+φzcos(φt)) \end{align} \begin{align} \dot{x}=\dot{y}cos (φt)-φy sin(φt)-\dot ...
0
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1answer
31 views

Let $f$ be differentiable $\exists\theta\backepsilon f(x+y)-f(x)=f '(x+\theta y)\cdot y$ Show that $\lim\limits_{y\rightarrow 0}\theta=\dfrac{1}{2}$

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be differentiable and such that $f''(x)\neq 0$. Then $\phi(\tau)=f(x+\tau y):\mathbb{R}\rightarrow\mathbb{R}$ is differentiable and $\phi '(\tau)=f'(x+\tau ...
0
votes
1answer
16 views

What is the mean of rate of change of a function with respect to the domain of some other function, as in: $\frac{dFz}{dy}$

While studying 'curl' I came accross these terms: Here, I don't understand the meaning of $\frac{dFz}{dy}$. Fz is a function of 'z', so what is the meaning of rate of change of Fz with respect to ...
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3answers
80 views

Finding the values of $\frac{dy}{dx}$ when $s = 2x^2+7x$ and $s = 0$.

A ball moves in the air according to equations $s = 2x^2+7x$ and $y = 3s^3-5s$. Find all values of $\frac{dy}{dx}$ when $s = 0$. I misinterpreted the question. I plugged the $s$ equation into ...
3
votes
0answers
55 views

gateaux derivative and frechet derivative

In calculus, we have the following equation $DF(x,y)=\partial F_xdx+\partial F_ydy$ if $F$ is differentiable. I think such equation still holds for frechet derivative, but not for gateaux derivative. ...
1
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1answer
26 views

Proving $f,g \in \mathbb D(U)$ (differentiable on $U$) $\implies f(x)g(x)$ is differentiable on U and $(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)$

This is done using compisition : $$x \mapsto^{F}(f(x),g(x))\mapsto^{B}f(x)g(x) \\ P(x)=B \circ F(x) \implies P'(x)h=B'(F(x))F'(x)h....(1)\\ \text{ I know that $B'(x)=B({}^1,\beta)+B(\alpha, ...
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1answer
50 views

help with chain rule

$$ \begin{align} D(4x+x^{-5})^{1/3}&=\left(\frac{1}{3}\right)\left(4x-x^{-5}\right)^{\frac{1}{3}-1}D\left(4x+x^{-5}\right)\\ ...
0
votes
1answer
84 views

Proof that applying the difference operator to a $d$-degree polynomial $d$ times yields $d!a_d$

Let $L$ be the lag operator and $\triangledown:=(1-L)$ be the difference operator, that is, given a polynomial $p(t)$, we have $$L(p(t))=p(t-1)\qquad \triangledown(p(t))=p(t) - p(t-1)$$ I am ...
3
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1answer
94 views

Solving linear differential equations

Find the general solution for the following equation: $$\frac{dy}{dt}+2ty=\sin(t)e^{-t^2}$$ Find a solution for which $y(0)=0$ First I found the integrating factor which is $e^{t^2}$ ...
12
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2answers
131 views

Prove that every such $f$ is $=0$ everywhere

Let $f: \mathbb{R} \to \mathbb{R}$ be differentiable; let $0 \leq f'(x) \leq f(x)$ for all $x \in \mathbb{R}$; and let $f$ vanish at some point. Prove that $f = 0$ on $\mathbb{R}$. Since ...
2
votes
6answers
219 views

How do I prove this $\frac{dx^n}{dx}=nx^{n-1}$ is true for every $n\geq 1$ to convince my students?

let $p_n(x)=x^n$ be a polynomial of degree $n$. I need help to be able to explain to my students why the derivative of $p$ is defined as follows: $$ p_n'(x)=\frac{dx^n}{dx}=nx^{n-1} $$ for every ...
2
votes
1answer
54 views

Show that Cauchy's function is infinitely differentiable

Show that $$f(x)= \begin{cases} exp(-\frac{1}{x^2}), & \text{if $x\gt 0$} \\[2ex] 0, & \text{if $x\le 0$ } \end{cases}$$ is infinitely differentiable. Clearly $f^{(n)}(x)=0$ for all $x\lt ...
1
vote
2answers
103 views

Is the function $\ln(ax + b)$ increasing/decreasing, concave/convex?

$h(x) = \ln(ax + b)$ NB. Examine your results acccording to values of $(a,b)$ I've differentiated twice in order to get the following: $$ h''(x) = -\frac{a^2}{(ax+b)^2} $$ I think this proves ...
1
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0answers
62 views

for $0<\alpha,\beta<2$, prove that $\int_0^4f(t)dt=2[\alpha f(\alpha)+\beta f(\beta)]$

I got the answer for the question but I have made an assumption, but I don,t know if it's correct. Attempt: Let $g(x)=\int_0^{x^2}f(t)dt$ and let $h(x)=g'(x)=2xf(x^2)$ now, applying intermediate ...
8
votes
3answers
199 views

Understanding implicit differentiation with concepts like “function” and “lambda abstraction.”

In high school, we learned to reason like so: $$(*) \qquad \frac{d}{dx}(x^2+x) = \frac{d}{dx}(x^2)+\frac{d}{dx}(x) = 2x+1$$ Now that I know more, I can "reanalyze" this chain of reasoning using ...
0
votes
1answer
58 views

Initial value problem, not sure where to begin!

Show that the function $y(t)=t^2$ satisfies the initial value problem $\frac{dy}{dt}=2\sqrt{y}, t\geq{0}; y(0)=0$ Show that this initial value problem does not have a unique solution, by ...
1
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1answer
26 views

How to set the variation of an integral to zero?

So I have an integral: $$\delta W = \int_{-\Delta}^\Delta \left[ x^2 \left(\frac{d\xi}{dx}\right)^2 - D_s\xi^2 \right] dx$$ Here $\xi$ is a function of $x$ and $D_s$ is a constant. $\Delta$ is just ...
0
votes
0answers
38 views

Using the mean value theorem assess: $\frac{|f(x,y,z)-f(0,0,0)|}{\|(x,y,z)\|}$ on a unit ball.

Using the mean value theorem assess: $\frac{|f(x,y,z)-f(0,0,0)|}{\|(x,y,z)\|}$ on a unit ball. $$f(x,y,z)=-(x^2+2y^2+3z^2-xy-2yz+xz+x+y+12)$$ To be frank, I don't understand what is asked of me ...
0
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0answers
100 views

Let $f$ be a twice differentiable fuction in $x$, and such that $f'$ is also differentiable in $x$. Show that $f''(x)=(f')'(x)$.

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a twice differentiable fuction in $x$, and such that $f'$ is also differentiable in $x$. Show that $f''(x)=(f')'(x)$. Let ...
1
vote
2answers
69 views

Find the derivative of $f(x) = \int_{-\infty}^\infty \frac{e^{-xy^2}}{1+y^2}\ dy.$

Problem statement: Find the derivative of $$f(x) = \int_{-\infty}^\infty \frac{e^{-xy^2}}{1+y^2}\ dy$$ and find an ordinary differential equation that $f$ solves. Find the solution to this ordinary ...
0
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1answer
29 views

Given that the function is of class $C^2$ prove the following.

Let $g:\mathbb{R} \to \mathbb{R}$ be of class $C^2$. Show that $$\lim_{h \rightarrow 0} \frac{g(a+h)-2g(a) +g(a-h)}{h^2} = g''(a)$$ How should one approach such questions? There are so many things ...
0
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0answers
40 views

Show that the function is of class $C^1$

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ such that $f(0,0)=0$ and $f(x,y)= \frac{xy(x^2-y^2)}{(x^2+y^2)}$, if $(x,y) \neq (0,0)$. Show that $f$ is of class $C^1(\mathbb{R}^2)$. If we use the ...
1
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1answer
71 views

Let $f$ be double differentiable function such that $|f′′(x)|\le 1$ for all $x\in [0,1]$. If $f(0)=f(1)$, then,

options: A) $|f(x)|>1 $ B) $|f(x)|<1 $ C) $|f′(x)|>1 $ D) $|f′(x)|<1$ attempt: I first tried using integration. $−1\le f′′(x)\le 1$ integrating from $0$ to $x$, $−x\le ...
1
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1answer
29 views

Derivative of $y^T(Ax)$

I'm not familiar with derivations of equations involving vectors and matrices. Given $$f(x)=c^Tx + y^TAx$$ with $y \in \mathbb{R}^d, A \in \mathbb{R}^{d\times n}, x \in \mathbb{R}^n, c \in ...
0
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1answer
39 views

Determine the maximum possible volume

A rectangular sheet of metal with dimensions 20 cm by 12 cm has squares removed from each of the four corners and the sides bent upwards to form an open box. Determine the maximum possible volume of ...
1
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2answers
77 views

$\cos(47^\circ)\sin(32^\circ)$ approximation by differentials

I need to approximate $\cos(47^\circ)\sin(32^\circ)$. In order to do this, I need to use differentials. So, for a function $f(x,y)$, we have: $$f(x,y)-f(x_0,y_0)\approx \frac{\partial ...
0
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1answer
53 views

Questioning the differentiability of $f(x,y)= x+ \frac{\sin y}{y}$ extended by $f(x,0)=x+1$

Determine the differentiability of $$f(x,y)=\begin{cases} x+ \frac{\sin y}{y}, & \text{if $y\neq 0;$ } \\x+1, & \text{if $y=0;$ } \end{cases}$$ I am using the Frechet derivative as my ...
1
vote
7answers
162 views

Finding the derivative of a function.

Differentiate $$f(x) = \sin(\ln(\cos(x^2+1)))$$ My work: $u = \ln(\cos(x^2+1))$ so $f(x) = \sin u$ , $f'(x) = \cos u = \cos(\ln(\cos(x^2+1)))$. I keep getting this answer, but where am I going ...
1
vote
1answer
92 views

Proving uniform continuity of function of two variables.

Proving uniform continuity of function:$$f(x,y)=\begin{cases} \frac{x^3-xy}{x^2+y^2}, & (x,y)\neq (0,0) \\ 0, & (x,y)=(0,0) \end{cases}$$ This is supposedly solve, but I don't understand the ...
0
votes
1answer
31 views

I'm asked to compute the gradient of a scalar function

$$h(x,y)=\begin{cases} y- \frac{\sin x}{x}, & x \neq 0; \\ y-1, & x=0 \end{cases} $$ So my thoughts are: $$\textrm{grad}(h(x,y))=\left(\dfrac{x\cos x-x \sin x}{x^2},1\right), \quad ...
0
votes
2answers
90 views

Gradient of a Frobenium norm cost Function

Folks - Please help. What's the gradient for the cost function below? $ D(Y||AX)=\frac{1}{2} ||Y-AX||^2_F $ Additional info - -need to get the derivative of that with respect to A. -Multiplicative ...
3
votes
3answers
100 views

Intersection of $36x^2 -9y^2+4z^2+36 = 0$ with plane $x=1$, derivative at a point

The exercise asks me to find the inclination of the line tangent to the intersection of $36x^2 -9y^2+4z^2+36 = 0$ with the plane $x=1$ in the point $(1,\sqrt{12},3)$, and then say to me that I have to ...
0
votes
1answer
53 views

Continuously Differentiable in $\mathbb{R^2}$

I understand the concept of continously differentiable (first derivative is continuous) in $\mathbb{R}$, however what does it mean for the RHS of: $\dfrac{d}{dt} ...
2
votes
3answers
105 views

Differentiability of this picewise function

$$f(x,y) = \left\{\begin{array}{cc} \frac{xy}{x^2+y^2} & (x,y)\neq(0,0) \\ f(x,y) = 0 & (x,y)=(0,0) \end{array}\right.$$ In order to verify if this function is differentiable, I tried to ...
0
votes
2answers
83 views

Finite difference differentiation formula

I'm trying to understand how the co-efficients of finite differences are calculated. In particular I'm interested in the first derivative for a uniform grid of unit width. I found this document ...
1
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0answers
61 views

We have $ f(x) = \sum_{n \geq 1} \frac{(x-1)^n}{n}$ prove that $f(x) = -\ln(2-x)$.

I am having problems with the following exercise, I have solved the first two parts of the exercise but I am unsure about the last part. I have the following power series $$f(x) = \sum_{n \geq 1} ...
-3
votes
2answers
55 views

Calculating derivatives applying chain rule,

Consider the functions f1(x)=2x+1, f2(x)=sin^2(x), f3(x)=ln(x). Calculate the first diffrentials of fi∘fj∘fk were {i,j,k} are all possible permutations of the numbers {1,2,3}. I calculated the first ...
-1
votes
3answers
82 views

Differentiate the following power series $\sum_{n \geq 1} \dfrac{(2x-2)^n}{n2^n+1}$

I am having issues with the differentiation of the following power series $$ \large f(x) = \sum_{n \geq 1} \dfrac{(2x-2)^n}{n2^n+1}$$ I get the following result $$ \large f'(x) = \sum_{n \geq 1} ...
2
votes
2answers
122 views

The sum of the lengths of the hypotenuse and another side of a right angled triangle is given.

The sum of the lengths of the hypotenuse and another side of a right angled triangle is given.The area of the triangle will be maximum if the angle between them is: ...