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

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2
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3answers
73 views

Lagrange multipliers from hell

I was asked to solve this question, decided to try and solve it with lagrange multipliers as I see no other way: "Find the closest and furthest points on the circle made from the intersection of the ...
2
votes
1answer
30 views

Showing that if derivative is 0, function is constant ($f: U \rightarrow \mathbb{R}$ where $U \subset \mathbb{R}^n$)

Here's the question: Suppose that $f: U \rightarrow \mathbb{R}$ is differentiable on the open subset $U\subset \mathbb{R}^n$, and $Df(x) =0$ for all $x\in U$. Show that $f$ is constant on $U$. My ...
0
votes
0answers
8 views

prove where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable and is not differentiable [duplicate]

Classify where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable -- where it's not, prove it, and where it is, prove it. For $x\neq 0$ and $y\neq 0$, we can just treat it as $f(a,b) = \sqrt{a+b}$ (where ...
0
votes
2answers
56 views

Show where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable

Classify where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable -- where it's not, prove it, and where it is, prove it. My thoughts: For $x\neq 0$ and $y\neq 0$, we can just treat it as $f(a,b) = ...
0
votes
1answer
24 views

How to find the Frechet differential of a functional?

We know that the Fréchet differential $DF(u,\delta)$ of a functional $F:V\to V$ is satisfied (cf. Wiki) $$ \lim_{\delta\to ...
2
votes
1answer
89 views

Second Derivative of Eigenvalue

Consider the potential $V_t=tx$ and the corresponding Schrödinger operator $H_t=- \frac{\partial ^2}{\partial x^2}+V_t$ on $L^2([0,R])$ with Neumann or Dirichlet boundary conditions. Let ...
1
vote
0answers
25 views

Weierstrass function

I got stuck on this exercise from Prof. Tao's real analysis notes. Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be the function $$f:= \sum_{n=1}^\infty 4^{-n} \sin(8^n\pi x)$$ Show that for every 8-dyadic ...
0
votes
1answer
19 views

applications of derivatives : maxima and minima

To finding the the maxima and minima why do we equate the derivative of a function with zero and n0t with any other number like 10,100 ?
2
votes
3answers
33 views

Derivative of sigmoid function

Sigmoid function is defined as $$\frac{1}{1+e^{-x}}$$ I tried to calculate the derivative and got $$\frac{e^{-x}}{(e^{-x}+1)^2}$$ Wolfram|Alpha however give me the same function but with exponents on ...
-4
votes
0answers
26 views

differentiate the given function. Simplify your answers [closed]

In Exercise 1 through 28, differentiate the given function. Simplify your answers y=√2X
2
votes
0answers
22 views

Total derivative

What is the significance and meaning of the total derivative? Why is it introduced in the definition of differentiability of scalar and vector fields?
1
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1answer
24 views

solving the derivative of a function with cos

my question is y=cos^4(2x^2-1) here is my work `Dy/dx=4cos^3(2x^2-1) d/dx cos(2x^2-1) Dy/dx=4cos^3(2x^2-1) (-d/dx(2x^2-1)sin(2x^2-1)) ...
0
votes
1answer
34 views

What is $p'(1-x), p(x)=x$?

Say if $p(x)=x $ and I want to find $p'(1-x)$ how do i go about it?. I would have thought it was $\frac{d}{d(1-x)}(x)$ but this doesn't give me the right answer.
0
votes
0answers
19 views

A question on differentiability and boundedness

Let $f:R\to R$ be a differentiable function such that limx->inf f'(x)=1. Show that f is unbounded. Here is my try For $\epsilon>0$, there exists an $M$ in reals such that $|f'(x)-1|<\epsilon$ ...
0
votes
1answer
57 views

Prove from the definition of differentiability that the function is differentiable at 2.

$$f(x) = \frac{x-1}{x+1}$$ From the Definition I have this so far. I am stuck and do not know how to continue. $$\begin{align} Q(h) &= \frac{f(h)-f(2)}{h} \\&= \frac{ \frac{h-1}{h+1} - ...
-1
votes
0answers
48 views

Differentiate from first principles $\sqrt{1+e^x}$ [duplicate]

Differentiate from first principles $\sqrt{1+e^x}$ i.e. $$\lim_{h\to0}\frac{f(x+h)-f(x)}h$$
2
votes
5answers
98 views

Derivative of $\; y={(1+e^x)}^{0.5}\; $ using the definition of the derivative

$$y={(1+e^x)}^{0.5} =f(x)$$ $$\frac{dy}{dx}= \lim_{h\to0}\frac {f(x+h)-f(x)}{h}$$ My attempt I got down to $$\lim_{h\to0}\frac{(1+e^xe^h)^{0.5}-(1+e^x)^{0.5}}{h}$$ I can't see where to go from ...
1
vote
3answers
22 views

A basic question on successive differentiation

How to prove that $$\frac{d^r}{dx^r}\cos x + i\frac{d^r}{dx^r}\sin x = i^r e^{ix}\ ?$$ I can understand it by putting values, but how to prove it?
-3
votes
0answers
35 views

Problem on differentiation and integration [closed]

there is a question that we want to know the answer of which described in the picture.
3
votes
2answers
136 views

Real analysis question involving a linear ODE

Where do I start with this one? This question is really quite difficult..
0
votes
1answer
22 views

Limits of Indeterminate Powers in Exponential Form using L'Hopital's Rule

I am trying to find the limit as $x \rightarrow 0$ of $x^x$ using L'Hopital's rule. I have written it in exponential form: $\lim\limits_{x \rightarrow 0} e^{x \ln x}$. I do not know how to put it in ...
0
votes
4answers
70 views

What is $\frac{1}{2} \int {e^{\frac{t}{2}}dt}$ equal to?

Would using substitution be helpful to get rid of the exponent of the variable? I tried substituting "$u$" in but it did not seem to help finding the integral.
1
vote
1answer
21 views

Single variable function derivative w.r.t. time?

I was studying calculus and I had doubts about this problem: (this is not homework) A circular wire expands due to heat so that its radius increases with a speed of $0.01 ms^{-1}$. How rapidly does ...
-2
votes
4answers
44 views

If $x^2 +xy =10$ then when $x=2$ what is $\frac{dy}{dx}$?

I solved for $y=3$ before I did the product rule and i'm not sure if that was the correct way to approach it.
0
votes
2answers
47 views

If $f(x)=x\sqrt{2x-3}$ what is $f'(x)$?

so far I re-wrote the problem using the product rule and chain rule so that i have $$\sqrt(2x-3)+x(\sqrt(2x-3)^{-1/2}$$ Now what?
0
votes
2answers
35 views

Find the parameter M

m(x+1)=e^|x| , m is a real number .Find the interval to which the parameter 'm' belongs , so that the previous equation has exactly two different solutions . Any idea how to approach this kind of ...
0
votes
0answers
18 views

Inverse functions determination by integral

From "Inverse functions and differentiation": Integrating this relationship gives $$ f^{-1}(x)=\int\frac{1}{f'(f^{-1}(x))}\,dx + c. $$ This is only useful if the integral exists. ...
2
votes
4answers
69 views

Second partial derivative test is inconclusive

I am trying to find the critical points of the function: $f(x,y)=2x^4-3x^2y+y^2$ and find the Max, Min and saddle points. What I've done so far is: $f_x=8x^3-6xy=0 , f_y=-3x^2+2y=0 , ...
0
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0answers
38 views
+50

Can't find gradient for MLE for mult-class logistic regression

$$P(k | x_i;w)= \frac{exp(w_k^tx_i)}{\sum_{j=1}^K exp(w_k^tx_i)}$$ $y_i^k$ is a vector that uses 1-of-k encoding. Thus, if $y_i=k$, then the vector $y_i$ has a 1 in the kth spot and a 0 everywhere ...
2
votes
1answer
28 views

Manipulating An Equation into A Workable Form

The question asks me to find the arc length of $$y= (x-x^2)^{1/2} + \sin^{-1}(x^{1/2})$$ I know I need to take the derivative: $$\frac{1-2x}{2(x-x^2)^{1/2}} + \frac{1}{(1-x)^{1/2}}$$ I've tried ...
1
vote
2answers
19 views

Application of chain rule

The equations $u=f(x,y),x=X(t),y=Y(t)$ define $u$ as a function of $t$, say $u=F(t)$. Compute $F'(t)$ in terms of $t$ if, $$f(x,y)=\log [(1+e^{x^2})/(1+e^{y^2})] , X(t)=e , Y(t)^t=e^{-t}.$$ From the ...
0
votes
0answers
15 views

Multivariable differentiation under the integral sign

Suppose that the functions $f:[a,b]$x$[a,b] \to \mathbb{R}$ and $\frac{\partial{f}}{\partial{t}}:[a,b]$x$[a,b] \to \mathbb{R}$ are continuous. Prove that the function $F:[a,b]$x$[a,b] \to \mathbb{R}$ ...
4
votes
0answers
24 views

Intuition on second order partial derivatives

Inspired by smooth submanifolds of $\mathbb{R}^n$, I am looking for a good geometric way to think of second order partial derivatives of a locally smooth function $f:\mathbb{R}^n \rightarrow ...
1
vote
4answers
59 views

How to differentiate $\frac{2x^5}{\tan x}$

$$\frac{2x^5}{\tan x}$$ I can differentiate $2x^5$ ($10x^4$) and $\tan x$ ($\sec^2 x$) but can't do that one Is there a rule I can apply?
0
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0answers
16 views

Nonexistence of a scalar field

Prove that there is no scalar field $f$ such that $f'(a;y)>0$ for a fixed vector $a$ and every non-zero vector $y$.
1
vote
1answer
25 views

Constant function on a convex set

If $f'(x;y)=0$ for every $x$ in an open convex set $S$ and every $y$ in $R^n$, prove that $f$ is constant on $S$. A set $S$ is called convex if for every $a$ and $b$ in $S$, ${ta+(1-t)b \epsilon S}.$ ...
1
vote
0answers
46 views

What exactly do the terms of $(f \circ g)'''$ mean?

Say, $g: X\to Y$ and $f: Y\to Z$ are smooth. One can find $(f \circ g)'''$ by using the Faà di Bruno's formula: $$(f \circ g)''' =(f'''\circ g)(g')^3 + 3(f''\circ g)g'g'' + (f'\circ g)g'''$$ But my ...
1
vote
0answers
18 views

Hessian matrix of $g\circ f$

Say, $f:\mathbb R^n\to\mathbb R^k$ and $g:\mathbb R^k\to\mathbb R$ are both $C^2$. I'd like to express the Hessian matrix of $g\circ f$ $$\left( \frac{\partial^2(g\circ f)}{\partial x_i \partial ...
0
votes
2answers
30 views

Leibniz's formula. [closed]

Let $f(x) =e^{3x}$. Differentiate $f(x)$ $n$ times and prove that $\sum_{k=0}^{n} 2^k \left ( \begin{array}{c} n \\ k \\ \end{array} \right ) = 3^n$. Hint: Use Leibniz's ...
1
vote
0answers
33 views

Total differentiation

For each of the functions below use the total diferential to approximate the change in $Y$ due to the given changes in $X$ and $Z$: $Y= X^2 + 4X -Z^2 -2XZ$, where $X=1$ and $Z = 4$ , and $\Delta X=2$ ...
0
votes
2answers
54 views

Why are the derivatives not treated the same?

It seems to me that derivatives are treated differently in certain places, but I do not understand why. Here is an example, if \begin{align} \frac{d}{dx} (\sqrt{1 + 4x^2}) & = \frac{1}{2\sqrt{1 ...
0
votes
3answers
59 views

How to find derivative of an integral

So I am given $$\frac{d}{dx}\left(\int_0^x e^{t^2}\ dt\right)$$ How would I go about solving this?
0
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1answer
18 views

Application of related rates

Here is a question from a sheet my math teacher assigned me. ...
0
votes
2answers
20 views

Maximize the directional derivative

Find the points $(x,y)$ and the directions for which the directional derivative of $f(x,y)=3x^2+y^2$ has its largest value, if $(x,y)$ is restricted to be on the circle $x^2+y^2=1$. For the point ...
1
vote
3answers
71 views

How to differentiate this

$$e^{\tfrac{1}{\sin x}}$$ Help me how to differentiate that please help me Thanks.
1
vote
1answer
41 views

Directional Derivative and differentiability

My question is similar, but not equal to this...Question on linearity of directional derivative Let $f'_{h}(a)$ be the directional derivative. And for the function $f:\mathbb{R}^n\rightarrow ...
5
votes
2answers
109 views

Prove $f $ is identically zero

$f:\Bbb R \to\Bbb R $ is differentiable, $f(0)=0$ and $|f'(x)|\le|f(x)|$ for all $x$ then prove $f$ is identically zero. I tried to use mean value theorem and end up in $|f(x)|\le |x||f(c)|$ for some ...
1
vote
1answer
67 views

When are $\Delta x$, $\delta x$, $dx$, and $\text{đ}x$ exactly the same? When are they approximately the same?

As a follow-up to this related question, I'd like to know under what circumstances, if any, $\Delta x$, $\delta x$ and $dx$ all mean the same thing, and under what circumstances they can all be said ...
0
votes
1answer
21 views

Derivative rule question

In this image,from a website on compound interest derivations, why are you allowed to take the derivative of JUST the limit? Shouldn't you have to take the derivative of the lefthand side of the ...
1
vote
1answer
28 views

$\frac{d(X'X)}{dX}=?$

Thanks a lot for reading my thread. I am wondering what is the derivative of $X'X$ with respect to $X$? Here $X$ is a vector/matrix, and $X'$ is the Hermitian matrix of $X$; It would be great if ...