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

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2answers
29 views

Stationary points of a function

$F(x)=x^3+Ax+B$ has a stationary point at $(-2,3)$. a) Find $A$ and $B$ and then find the nature of all stationary points. Thank you!
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1answer
58 views

Weierstrass Caratheodory on open interval

I have been working on this question for a while now, and if I have understood it correctly shouldn't the answer be that $\phi_{c}=f'(x)$ for all $x \in (a,b)$ as the function f , is now said to be ...
1
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0answers
25 views

Derivatives of function defined implicitly

The two equations $F(x,y,u,v)=0$ and $G(x,y,u,v)=0$ determine $x$ and $y$ implicitly as functions of $u$ and $v$, say $x=X(u,v)$ and $y=Y(u,v)$. Show that $$\frac{\partial X}{\partial ...
4
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1answer
71 views

Do rational and irrational numbers flip-flop?

I have found out that between every 2 rational numbers there is an irrational number, and between every 2 irrational numbers, there is a rational number. Does this mean that the rational and ...
3
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3answers
130 views

Finding a tangent to an ellipse parallel to a given line

Problem: Find the lines that are tangent to the ellipse $x^2 + 4y^2 = 8$ and parallel to $x +2y = 6$. I tried to find the derivative of $x^2 + 4y^2 = 8$ and I got: $$\frac{dx}{dy} = -\frac{x}{2y}.$$ ...
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1answer
30 views

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 ...
11
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7answers
1k views

What's wrong with these equations? [duplicate]

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
27 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 ...
0
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1answer
42 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
56 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?
12
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2answers
111 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
29 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' ...
0
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1answer
27 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 ...
0
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1answer
26 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 ...
1
vote
1answer
116 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
87 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
63 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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1answer
139 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 ...
0
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1answer
27 views

Derivative of a power series

Suppose the sequence $(b_k) , k\geq 0$ satisfies $\sum k|b_k| < \infty$, then show that $\sum_{k=0}^\infty b_kx^k$ converges uniformly to a function $g$ on $|x| \leq 1$ and that $g'(x) = ...
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1answer
165 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 ...
2
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1answer
46 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 ...
0
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1answer
68 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
31 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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2answers
24 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?
2
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1answer
53 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 ...
2
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1answer
73 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) = ...
2
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3answers
104 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
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1answer
46 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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0answers
10 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
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2answers
88 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) = ...
1
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1answer
42 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
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1answer
124 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 ...
2
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1answer
55 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
25 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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3answers
65 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 ...
2
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0answers
35 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
32 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
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1answer
43 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
21 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
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1answer
77 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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5answers
110 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
42 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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2answers
154 views

Real analysis question involving a linear ODE

Where do I start with this one? This question is really quite difficult..
0
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1answer
65 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
80 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
27 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
74 views

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

If $x^2 +xy =10$ then when $x=2$ what is $\frac{\mathrm dy}{\mathrm dx}$? 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
73 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
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2answers
41 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
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2answers
40 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. ...