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

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Derivative of Normal Vector Field

This is an example from Do Carmo (Example 4, page 139). Consider the saddle point $p=(0, 0, 0)$ of the hyperbolic paraboloid $z=y^2-x^2$ with parameterization $\mathbf x(u, v)=(u, v, v^2-u^2)$. It is ...
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4answers
92 views

What's wrong with this equations?

My friend Boris (Boryan) gave me a task, and completely refuses to give the answer what's wrong here. $$x^2=\overbrace{x+\cdots+x} ^{x\text{ times}}$$ $$(x^2)'=(x+\cdots+x)'$$ $$2x=1+\cdots+1$$ ...
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1answer
15 views

Tangent plane that passes through a point

How would I find the (a,b) that satisfies that the tangent plane to $f(x,y) = (x^2) + 2xy + (y^2)$ passes through the point $(2,1,0)$ ? I know that $f(x)= 2x + 2y$, and $F(y): 2x + 2y$. Therefore ...
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0answers
10 views

Partial derivative of a Piecewise function with two variables

I'm having trouble solving partial derivatives of a piecewise function where the function breaks. For the following equation, what would the partial derivatives (both x and y) be? f(x,y): {x if ...
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1answer
31 views

Partial derivative of a Piecewise function

If I have the following equation: $$ f(x,y) = \begin{cases} x; & y \ge 0 \\ y; & y < 0 \\ \end{cases} $$ What are the partial derivatives (both x and y) of the function? I have trouble ...
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1answer
47 views

Derivitative of $\sqrt[3]{6x + 3}$

Today I was learning with the wolframalpha problem generator and I got the following exercise Is this a mistake? How did they get to this solution?
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2answers
78 views

Second derivative of $f(f(\cdots f(x)\cdots )?$

For convenience, let's write $f_n(x)=f(f(\cdots f(x)\cdots )$ where $f$ is iterated $n$ times. Suppose: $$f(0)=0,\quad f'(0)=\alpha,\quad f''(0)=\beta$$ What is $f''_n(0)?$ I've found ...
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1answer
18 views

consequence of Mean Value Theorem

Let $f$ a continuous function on $[a, b]$ $a < b$ ,derivable on $(a, b)$ then there exist $c_1, c_2 \in (a, b)$ ,$c_1 \ne c_2$ such that $\frac{f (b) - f (a)}{b - a} = \frac{f '(c1) + f' ...
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1answer
9 views

Convex Subset Projection

Suppose that C is a closed convex subset of $\mathbb R^n$ and $x \in \mathbb R^n$. The projection of $\mathbf x$ onto C is the closest point $\mathbf y \in C : \mathbf z = \mathbf y$ minimizes ...
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1answer
21 views

Prove that $f$ is differentiable on $\Bbb R$ and find the derivative.

$$f(x) = g(x)|g(x)|$$. I know that to prove that a function is differentiable, I need to prove that $$\lim_{x \to c} \frac {f(x) - f(c)}{x-c}.$$ And then to prove that the function is ...
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1answer
36 views

Proving the chain rule by first principles

I'm currently trying to prove: $(f(g))'(a)=f'(g(a))*g'(a)$ I have been given a proof which manipulates: $f(a+h)=f(a)+f'(a)h+O(h)$ where $O(h)$ is the error function. However, I would like to have a ...
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4answers
78 views

Why does the result follow?

How does this theorem follow? Theorem. If $g$ is differentiable at $a$ and $g(a) \neq 0$, then $\phi = 1/g$ is also differentiable at $a$, and $$\phi'(a) = (1/g)'(a) = -\frac{g'(a)}{[g(a)]^2}.$$ ...
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2answers
37 views

n'th and (n-1)'th derivative of $\frac{1}{x^n - a}$

I have a function of the form $f(x) = \frac{1}{x^n - a}$, and I need to programmatically find the n'th and (n-1)'th derivative of the function. Since the function has this specific form and that the ...
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0answers
30 views

differentiating with different variables [on hold]

How can I differentiate this question which has different variables of differentiation? $$\frac{\partial y}{\partial t}( ky(1- \frac {y}{L}))$$ Where $k ,L$ are constants.
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1answer
27 views

Calculus - Trig Maximum Value Problem

When the rules of hockey were developed, Canada did not use the metric system. Thus, the distance between the goal posts was designated to be six feet. If Sidney Crosby is on the goal line, three feet ...
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1answer
94 views

Are “most” continuous functions also differentiable?

Let $A$ be a nonempty open subset of $\mathbb{R}$. Consider a function $f : A \rightarrow \mathbb{R}$. Given that $f$ is continuous, what is the probability that it is differentiable? I suspect it ...
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1answer
13 views

Tangent planes perpendicular at each point of intersection

Find the set of all points $(a,b,c)$ in 3-space for which the two spheres $(x-a)^2+(y-b)^2+(z-c)^2=1$ and $x^2+y^2+z^2=1$ intersect orthogonally.( Their tangent planes should be perpendicular at each ...
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0answers
8 views

Multi-variables piecewise function

Given the function defined by the rule: $$ f(x,y)= \left\{ \begin{array}{ll} x(1-y) & \qquad \mathrm{if}\quad x\leq y \\ y(1-x) & \qquad \mathrm{otherwise} \\ ...
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0answers
14 views

Volume element notation

If $X=(x_1,x_2,\cdots,x_n)$. The notation $dX$ usually means $$dX:=(dx_1,dx_2,\cdots,dx_n)$$ or $$dX:=dx_1dx_2\cdots dx_n?$$ I am little bit confused about this, can anyone explain this for me? ...
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1answer
53 views

What is the exact meaning of Differentiability?

What is the exact meaning of Differentiability of a function at a given point? I know that If $\lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}$ exists, then we can say that the given function $f(x)$ is ...
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1answer
24 views

Differentiability problem .

Hi can someone help me with the following problem. I am having difficulties evaluating : $$ \frac {d} {dt} f'(u(t)) $$ Is it just $f''(u(t))$ ? Thanks
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19 views

finding the equation of tangent to curve involving cos

I need help getting the equation of the tangent to the curve $y=3 \csc(2x)$ at $x=3\pi/4$. I used WA and got the derivative as $y'=-6\cot(2x)\csc(2x)$ I need to know how to get the derivative and ...
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2answers
21 views

implicit differentiating equation with $\cos$

I need help getting $\frac{d^2y}{dx^2}$ for $y−\cos y=2x$ Someone answered and got $(1+\sin y(x))3+4\cos y(x)$ but i was unable to follow their steps and didnt get how to do it. any HELP?
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2answers
35 views

Derivative of a Matrix to a Power

Fix a positive interger $k$ and let $F: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times n}$ be the map on $n \times n$ matrices defined by $F(A)= A^k$. Show that $F$ is differentiable at ...
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1answer
59 views

Prove where $|x|^2(\sin(\pi|x|))^2$ (piecewise) is differentiable in $\mathbb{R}^2$

List all points in $\mathbb{R}^2$ at which $f$ is differentiable as well as ALL points in $\mathbb{R}^2$ where $f$ is not differentiable (implied by the first list) when \begin{equation} f(x) = ...
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3answers
64 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 ...
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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 ...
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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 ...
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2answers
54 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) = ...
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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 ...
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1answer
71 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 ...
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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 ...
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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 ?
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2answers
18 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 ...
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26 views

differentiate the given function. Simplify your answers [on hold]

In Exercise 1 through 28, differentiate the given function. Simplify your answers y=√2X
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0answers
21 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?
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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)) ...
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1answer
33 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.
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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 $|f'(x)-1|<\epsilon$ ...
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1answer
50 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} - ...
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Differentiate from first principles $\sqrt{1+e^x}$ [duplicate]

Differentiate from first principles $\sqrt{1+e^x}$ i.e. $$\lim_{h\to0}\frac{f(x+h)-f(x)}h$$
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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 ...
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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?
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Problem on differentiation and integration [on hold]

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

Real analysis question involving a linear ODE

Where do I start with this one? This question is really quite difficult..
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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 ...
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4answers
68 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.
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1answer
20 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 ...
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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.
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2answers
45 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?