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

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Derivatives of functions defined implicitly

Let $f$ and $g$ be functions of one real variable and define $F(x,y)=f[x+g(y)]$. Find the formulas for the partial derivatives of $F$ of first and second order, expressed in terms of derivatives of ...
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12 views

Find derivation error and how to prevent it in the future

Problem: Here Calculation: Here. In the yellow spot, I get it wrong. Then I correct it and arrive at the right answer. Question: I'd be glad if someone could find my derivation error and advice me ...
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0answers
26 views

True or false: differentiation. [on hold]

If the function $f(x,y): \mathbb{R}^2 \longrightarrow \mathbb{R}^3$ is differentiable at $(2,-1)$ with a tangent plane such as $z= 2x - 3y + 2$, then the function $g(x,y)= 3x - 2f(x,y) + 5$ is ...
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4answers
506 views

Is it necessary that every function is a derivative of some function?

I thought about this a lot and consulted a lot of people but everyone had contradicting answers. I am a high school student. please help.
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1answer
27 views

Differentiability of $\sum x^j$

Prove that $\sum x^j$ is differentiable on (-1,1), and $$\frac{d}{dx} \sum x^j = \sum (j+1) x^j$$ I am able to prove that $\sum x^j$ converges uniformly to $\frac{1}{1+x}$. However, how do I get this ...
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2answers
50 views

Find the derivative of $1/\sqrt{1+x^2-\cos^2x-e^{2\pi \cos(\sin 1/x)}}$

(calculus) How can I prove that $$\frac{d}{dx}\frac{1}{\sqrt{1+x^2-\cos^2x-e^{2\pi \cos(\sin 1/x)}}}=\frac{-\frac{\displaystyle\pi\sin(\sin(1/x))\cos(1/x)e^{2\pi\cos(\sin(1/x))}}{x^2}+x+\sin x+\cos ...
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1answer
16 views

Differentiability in two variables - directional derivative & gradient

I have read a chapter about differentiability in two variables. I now have two questions: Why do we need the constraint that $|\vec{u}|=1$ when we calculate the directional derivative? Definition of ...
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1answer
23 views

Differentiation of $u(t)=\int_0^t h(s,t)ds, \ \forall t \in \mathbb{R}$ with the multivariable chain rule

Problem: Let $h: \mathbb{R}^2 \to \mathbb{R}$ be continuous and differentiable with respect to its second variable, define $u(t)= \displaystyle \int_0^t h(s,t)ds, \ t \in \mathbb{R}$ In an ...
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1answer
40 views

Uncomfortable using Leibniz notation for the chain rule.

I am working through the following solved problem which uses separation of variables to get two ODEs. The problem is to show that $$\frac{1}{\sin\theta ...
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2answers
18 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
33 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 ...
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0answers
20 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 ...
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1answer
43 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 ...
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3answers
33 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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3answers
25 views

Find all points on the curve $y=2x+x^{-1}$ which have a tangent parallel to the x-axis

Find all the points on the curve $y=2x+x^{-1}$ which have a tangent parallel to the $x$-axis.
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1answer
27 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 ...
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984 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
20 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
11 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
49 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
87 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
22 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
39 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
82 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
39 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
28 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
106 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
17 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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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
54 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
26 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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0answers
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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
44 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
70 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
55 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
87 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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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 ...
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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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3answers
31 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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differentiate the given function. Simplify your answers [closed]

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