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

learn more… | top users | synonyms (2)

12
votes
2answers
119 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
162 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
37 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
48 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
vote
0answers
42 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 ...
6
votes
2answers
140 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
52 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
vote
1answer
20 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
36 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
votes
0answers
85 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
66 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
votes
1answer
26 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
votes
0answers
30 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 ...
2
votes
1answer
50 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 ...
0
votes
1answer
26 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
votes
1answer
24 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
vote
2answers
55 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
votes
1answer
44 views

Questioning 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}$

$$f(x,y)=\begin{cases} x+ \frac{\sin y}{y}, & \text{if $y\neq 0;$ } \\x+1, & \text{if $y=0;$ } \end{cases}$$ I am using the Frechet derivative as my definition of differentiability. Since ...
1
vote
7answers
129 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 ...
2
votes
1answer
43 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
30 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
60 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
81 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
48 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
89 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
1answer
53 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
vote
0answers
56 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} ...
-5
votes
2answers
45 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
49 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
39 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: ...
2
votes
1answer
54 views

The least value of the function $f(x)=|x-a|+|x-b|+|x-c|+|x-d|$ [duplicate]

If $a<b<c<d$ and $x\in\mathbb R$ then what is the least value of the function $$f(x)=|x-a|+|x-b|+|x-c|+|x-d|\ ?$$ $f(x)= \begin{cases} a-x+b-x+c-x+d-x & x\leq a \\ ...
1
vote
0answers
41 views

differentiating a smooth function defined by an integral

Suppose we define a function by the integral $$ f(x) = \int_{-\infty}^{\infty}g(x,y) dy $$Suppose we know that $f(x)$ is smooth. Does this mean that necessarily $$ ...
0
votes
0answers
17 views

Finding the $h'(x,y,z)$ if $h= p \circ q $ $p(x,y,z)=(x \sin y, x \cos y, z+y ), q(x,y,z)=(x^2,x+y,2e^z)$

I just want someone to check my work basically. Providing thoughts and insight, into possible mistakes: Finding the $$h'(x,y,z)$$ if $$h= p \circ q ,\ \ p(x,y,z)=(x \sin y, x \cos y, z+y ), \ \ ...
1
vote
1answer
60 views

Find the derivative of the following by definition: $f(x,y)=(x^3, xy^2-y^2)$

$$f(x,y)=(x^3, xy^2-y^2)$$ So with these types of functions the derivative is $f'(x,y)=\begin{pmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ ...
0
votes
1answer
48 views

Finding the partial derivatives of $f(x,y,z)= \int_{0}^{x}t^tdt + x ^{\sin(y^z)}$ and the first derivative.

$$f(x,y,z)= \int_{0}^{x}t^tdt + x ^{\sin(y^z)}$$ The derivative would be $f'(x,y,z)(h^1,h^2,h^3)= \frac{\partial f}{\partial x}h^1+\frac{\partial f}{\partial y}h^2+\frac{\partial f}{\partial z}h^3.$ ...
1
vote
1answer
46 views

How would I find the second derivative of the bilinear $B(x,y)=Ax \times y$ where $A=\begin{pmatrix} 1 & 2&3 \\ 0 & -1 & 2\\ -1 & 2 & 4 \end{pmatrix}$

$$B(x,y)=Ax \times y \text{ where } A=\begin{pmatrix} 1 & 2&3 \\ 0 & -1 & 2\\ -1 & 2 & 4 \end{pmatrix}$$ Second derivative is obviously the first derivative of the first ...
3
votes
1answer
124 views

Find the $n^{th}$ derivative of $\frac{x^n}{(1+x)}$

Find the $n^{th}$ derivative of $\frac{x^n}{(1+x)}$ . I think we have to use Leibnitz's Formula to evaluate this, but I haven't succeeded in it as well. I have already received an answer of $\frac ...
1
vote
1answer
36 views

If all derivatives are zero at a point, what does this imply?

Let's say I have a function $f$ which for all positive $n$ and some complex point $z_0$ satisfies $f^{(n)}(z_0) = 0$. What does this say about the function's analyticity or holomorphicity? Obviously, ...
2
votes
1answer
58 views

Proofs that Dirichlet's function is not differentiable

Define $f: (0,1) \to (0,1)$ by $f(x)= \begin{cases} \frac{1}{q}, & \text{if $x=\frac{p}{q}$ in lowest terms with $p,q \in \mathbb{N}$} \\ 0, & \text{if $x$ is irrational} \end{cases} $ The ...
0
votes
0answers
33 views

Differentiability proof of two variable function

Show that the function is differentiable: $$f\left(x,y\right) = \begin{cases} \frac{1}{y} \sin \left( xy \right) &\mbox{if } y \ne 0 \\ x & \mbox{if } y =0. \end{cases} $$ I know the general ...
3
votes
1answer
127 views

Integral of $2^{2^{2^x}}$?

$$\int2^{2^{2^x}}~\mathrm{d}x$$ Derivative is $\ln^3(2)2^{2^x+x+2^{2^x}}$. So no substitution technique can be used. So please guide, I am confused. Is this elliptic?
1
vote
1answer
27 views

Prove that all terms of a sequence of functions are convex.

Let $\ f_{n}: [0,1] \rightarrow \mathbb R, \quad f_{n}(x) = \left(e^{x}\right)^{1/n}.$ Is there a natural $n$ such that $f_{n}$ is concave on $[0,1]$? So second derivative is ...
2
votes
3answers
274 views

Finding the derivative of an absolute value

This one I just don't know how to derive. $\ln\|x^4cosx||$ I know the derivative of $\ln\ x$, is just $\frac{1}{x}$ . It is the absolute value that throws me off. My question is, does the absolute ...
2
votes
2answers
55 views

Show $\frac{d^2x}{dt^2}=(1+\cos x)(x+\sin x)$

Show $\dfrac{d^2x}{dt^2}=(1+\cos x)(x+\sin x)$ given $\dfrac{dx}{dt}=x+\sin x$. Thought it would just be $(1+\cos x)$.
3
votes
2answers
86 views

Derivative of $f(3x+1,3x-1)=4$

This exercise asks me to take the derivative of $$f(3x+1,3x-1)=4$$ where this equality is said to be valid for all $x$. The exercise specifically asks me to prove that ...
0
votes
1answer
24 views

Derivative of double integrals with respect to one or more upper limit(s)

I'd like to make sure I'm performing the following correctly: $\frac{d}{db} \int_0^\bar v \int_0^b h(v).g(r) \;dr\; dv + \frac{d}{db} \int_0^b \int_0^\bar r h(v).g(r) \;dr\; dv - \frac{d}{db} ...
1
vote
1answer
29 views

One-sided derivative of composition function

$f : V \subset\mathbb R^n \to \mathbb R$ is differentiable, $g : [0,1] \to V$ a continuous function. Given $g(1)=p, Df(p)=0$, and that $f\circ g $ is left differentiable, can we deduce that the left ...
1
vote
1answer
51 views

Is differentiation of zero, zero?

I was just thinking about the question and googled it but couldn't get anything, is it zero because its a constant function or it is anything more complicated??
0
votes
0answers
20 views

Describing set of points where a convex function is differentiable

I've been told that the set of points at which a convex function $f: \mathbb R^n\rightarrow \mathbb R$ is differentiable is an $F_{\sigma}$ set, and I was hoping someone could help me see this. ...
0
votes
2answers
45 views

Supremum and infimum of $\left\{(n^2+2n+1)^{\frac{1}{n^2}} \mid n \in\mathbb N \right\}$

Task is to find infimum and supremum of $\left\{(n^2+2n+1)^{\frac{1}{n^2}} \mid n \in\mathbb N \right\}.$ I start from calculating derivative of $ f:\mathbb{R} \rightarrow \mathbb{R}$ where $ ...