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

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1answer
43 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
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7answers
125 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
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1answer
38 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 ...
1
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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 ...
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2answers
58 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 ...
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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 ...
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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
86 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 ...
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1answer
49 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 ...
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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} ...
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2answers
43 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 ...
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3answers
36 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
35 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
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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
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0answers
39 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
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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 ), \ \ ...
3
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1answer
59 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} \\ ...
3
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1answer
35 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 $f'(x,y,z)(h^1,h^2,h^3)= \frac{\partial f}{\partial x}h^1+\frac{\partial f}{\partial y}h^2+\frac{\partial f}{\partial z}h^3.$ ...
3
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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
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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
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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
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0answers
30 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
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1answer
126 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
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1answer
23 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
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3answers
272 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
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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
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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 ...
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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
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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 ...
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1answer
47 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??
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0answers
19 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. ...
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1answer
31 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 $ ...
2
votes
2answers
57 views

Difference in definition of differentiation

Okay so quite often I see two different definitions of differentiation and I want to know when it is appropriate to use each one. $$\lim_{h \to 0} \frac{f(x_{0}+h)-f(x_{0})}{h}$$ and $$\lim_{x \to ...
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0answers
16 views

more dimensional derivatives

Let variable $x \in \mathbb{R}^n$ and let $\theta \in \mathbb{R}^n$ be a constant. We have 2 other variables $s \in \mathbb{R}$ and $y \in \mathbb{R}^n$ such that $x = s\theta + y$. In my course ...
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2answers
53 views

differentiation of 1-norm of a vector

Assume you want to find the minimum of the following expression $\|x\|_1 + \alpha \|Ax-b\|_2$ where $x\in R^N$. So basically I want to calculate the derivative of $\|x\|_1$ so I could finally set ...
3
votes
3answers
113 views

What is the derivative of $|x^3|$?

Let $f(x)=|x^3|$. I found two ways to differentiate this function. Apparently method 2 is wrong, but I cannot figure out why. So the question is, is method two wrong and why? Method 1 (according to ...
3
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4answers
46 views

Show that $f$ is a decreasing function

It's given that $f(x)=\frac{1}{x^3}-x^3$ for $x>0$ show that $f$ is a decreasing function. My attempt $f'(x)=-3x^{-4}-3x^2$ $x^6=-1$ How to continue by my attempt ?
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1answer
6 views

Covariant derivative and tensor symmetries

Suppose we have a tensor field $T^{ab}$ such that $T^{ab} = T^{ba}$ everywhere. Then from the definition of the Riemannian covariant derivative in terms of a map between tensors, why must we then have ...
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1answer
61 views

Proof of a known theorem ($f_{xy}=f_{yx}$)

It's a pretty famous result but I'm not sure how to prove it. Let $ f: A \subset \Bbb{R^2} \to \Bbb{R} $ and $(a,b)\in A$ such that $f_{xy}$ and $f_{yx}$ exist in neighbourhood of $(a,b)$ and ...
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0answers
17 views

Differentiability of functions [duplicate]

What is the definition of a differentiable function and the theorems governing differentiability?
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1answer
29 views

Uniform bound on the error from the derivative

Suppose $f: \mathbb{R}^m \to \mathbb{R}^n$ is differentiable. For each $x$, the function $$h \mapsto f(x+h) - f(x) - Df_x(h)$$ is $o(|h|)$ at $h=0$. Assuming $f$ is $C^1$ (or more, if we need), I'm ...
5
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1answer
42 views

L'Hôpital's rule and Difference Quotients

Consider the general difference quotient for a function $f(x)$ that is differentiable at $x = a$: $$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$ Since both the numerator and denominator of the ...
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0answers
25 views

Symmetry of the second derivative

For the purposes of this question, a function $f$ is differentiable at $x\in \mathbb{R}^d$ iff (i) the directional derivative $\mathrm{D}_vf(x)$ exists for all $v\in \mathrm{T}_x(\mathbb{R}^d)$ and ...
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1answer
44 views

Differentiation under integral sign.

There is a integral that has to be differentiated. In an article i studied that the derivative of the integral of a function is the integral of the derivative of that function with respect to some ...
2
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2answers
30 views

Differentiability and continuity of a multivariable function

Let $f:\mathbb R^2\to \mathbb R$ be defined by $$f(x,y)=\begin{cases}\frac{x|y|}{\sqrt{x^2+y^2}},& (x,y)\ne(0,0)\\ 0,& (x,y)=(0,0).\end{cases}$$ For which non-zero vectors $u$ does ...
0
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0answers
27 views

Classifying Critical Points of $f(x,y)=xy-x+2x^3-yx^3$

I am classifying the critical point(s) of $ f(x,y)=xy-x+2x^3-yx^3 $: I first found the critical points by solving for $ f_x=f_y=0 $: $f_x= y-1+6x^2-3yx^2=0 $ $f_y= x-x^3=0$ Hence $x=0$ and ...
1
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1answer
30 views

Derivative of polynomial in GF(16)

how can I find the derivative of the following polynomial in $GF(2^4)$: $\alpha x^4+x^3+\alpha x^2+\alpha^2 x+1$ ?
0
votes
1answer
24 views

Stationary points of piecewise function

I'm studying a piecewise function: $$ y = f(x) = \begin{cases} 0 & \text{if} \quad x \geq 0 \\ 1 - \sqrt{1 - x^2} & \text{otherwise} \end{cases} $$ The first derivative is: $$ f'(x) = ...
0
votes
1answer
39 views

Writing an expression for a change in angular velocity of an angle

Let $AB$ is rotating at $\omega_{AB}=4$ rad/s. Find $\omega_{CD}$ when $\theta=\pi/6$. So the first thing I did was wrote an express for $CD$ call it $r$. $\phi$ is Angle $CAB$ for reference. By ...