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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0answers
53 views

Interchange of limiting operations (question from an engineer)

I need to clarify when are the below operations valid. If possible, please link me to the related theorems, where I can find details. 1- Given a double integral \begin{equation} \int_{X}\int_Y ...
0
votes
3answers
47 views

Proving Differentiability rigorously

Assume that real function f is differentiable at $x_0$ with $f'(x_0)$ >0. How would one show that there exists a $\delta$>0 such that $$ f(x)>f(x_0) $$ for all x in between $x_0$ and $ x_0 + ...
4
votes
1answer
54 views

Differentiability implies continuity — possibly pedantic question about the common proof

The common proof that differentiability implies continuity arrives at this limit: $$\lim_{x\to a} [f(x) - f(a)] = 0$$ I'm failing to see the simple justification for moving to the next step, which ...
1
vote
1answer
48 views

One-sided Derivative Question

Let's say we define $$D_{+}f(x):=\lim_{h\to 0^+}\frac{f(x+2h)-f(x+h)}{h}$$ to be the "right-handed" derivative. This way the function does not have to exist (or equal what it 'should') at the point ...
1
vote
1answer
40 views

Let P(x) be a polynomial of degree 4 , having extremum at $x=1,x=2$ and $\lim_{x\to 0}\frac{x^2+P(x)}{x^2}=2$ Then the value of P(2)

Let P(x) be a polynomial of degree 4 , having extremum at $x=1,x=2$ and $\lim_{x\to 0}\frac{x^2+P(x)}{x^2}=2$ Then the value of P(2) is _______ I worked out the limit using L'Hospital got a relation ...
2
votes
3answers
221 views

What is defined by rate of change at a single point?

Rate of change measures how fast a process is going when it moves from one point to another. It measures the change of, say, $Y$ when $X$ moves from $X$ to $X + \Delta X$. But my problem arises when ...
0
votes
3answers
54 views

Where did the linear approximation/linearization formula come from?

Where did the linear approximation/linearization formula come from? I understand that it takes root in the point-slope form and slope intercept form of a linear equation, but I don't understand where ...
0
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0answers
28 views

Intermediate value of the derivative.

Hi all what would the best way be to approach this question? I tried using the hint but I can't seem to formulate an answer for the fist part. Any help for the first and second parts of the question ...
1
vote
1answer
35 views

Differentiation of $u^{T}Su$

I want to differentiate $u^{T}Su$ wrt $u$ where $u$ is $n$ x $1$ and $S$ is $n$ x $n$matrix . So I did the following . Since $u^{T}Su$ is a number , I wrote its expression ie $$ f = ...
0
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0answers
9 views

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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0answers
14 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 ...
30
votes
4answers
2k 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.
1
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1answer
29 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 ...
2
votes
2answers
80 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 ...
1
vote
1answer
30 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 ...
3
votes
1answer
41 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 ...
4
votes
1answer
65 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 ...
0
votes
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!
0
votes
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
vote
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
votes
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
votes
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: $$\frac{dx}{dy} = -\frac{x}{2y}.$$ ...
1
vote
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
votes
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. $$x^2=\overbrace{x+\cdots+x} ^{x\text{ times}}$$ $$(x^2)'=(x+\cdots+x)'$$ $$2x=1+\cdots+1$$ ...
0
votes
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
votes
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 ...
0
votes
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
votes
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 ...
1
vote
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
votes
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
votes
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
119 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 ...
1
vote
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 $$\phi'(a) = (1/g)'(a) = -\frac{g'(a)}{[g(a)]^2}.$$ ...
0
votes
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 ...
1
vote
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) = ...
5
votes
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
votes
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
votes
1answer
70 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 ...
0
votes
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
votes
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
votes
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
votes
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
votes
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 ...
0
votes
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
votes
2answers
89 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
vote
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
votes
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
votes
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 ...