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

learn more… | top users | synonyms (2)

1
vote
0answers
5 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 ...
1
vote
0answers
7 views

Higher Order Derivative Proof .

I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ ...
0
votes
2answers
19 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 + ...
2
votes
1answer
28 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 ...
4
votes
1answer
103 views

A tough one: show that this is not differentiable at any point in R

Here's the question: Define $\phi: \ \mathbb{R} \rightarrow \mathbb{R}$ by $$ \phi(x) = \begin{cases}x & 0\leq x\leq\frac{1}{2}\\ 1-x & \frac{1}{2}\leq x\leq 1\end{cases}. $$ And then ...
1
vote
1answer
32 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 ...
2
votes
3answers
139 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 ...
1
vote
1answer
15 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 ...
0
votes
3answers
37 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 ...
26
votes
4answers
1k 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.
0
votes
0answers
24 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
31 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
votes
0answers
8 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 ...
0
votes
1answer
37 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 ...
3
votes
1answer
210 views

Derivation of Euler-Lagrange equation

Here is a simple (probably trivial) step in the derivation of the Euler-Lagrange equation. If we denote $Y(x) = y(x) + \epsilon \eta(x) $, I want to know why is $\dfrac{\partial ...
0
votes
0answers
13 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 ...
0
votes
0answers
28 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 ...
2
votes
2answers
191 views

Show that if $f(x)= \sum\limits_{i=0}^n a_i x^i$ and $a_0+\frac{a_1}{2}+\ldots+\frac{a_n}{n+1}=0$, then there is an $x \in (0,1)$ with $f(x)=0$

Show that if $f(x)= \sum\limits_{i=0}^n a_i x^i$ and $a_0+\dfrac{a_1}{2}+\ldots+\dfrac{a_n}{n+1}=0$, then there is an $x \in (0,1)$ with $f(x)=0.$
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 ...
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
2answers
51 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
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 ...
1
vote
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 ...
3
votes
1answer
29 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 ...
11
votes
7answers
994 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. $$x^2=\overbrace{x+\cdots+x} ^{x\text{ times}}$$ $$(x^2)'=(x+\cdots+x)'$$ $$2x=1+\cdots+1$$ ...
4
votes
1answer
41 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
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!
1
vote
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 ...
3
votes
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 ...
3
votes
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: $$\frac{dx}{dy} = -\frac{x}{2y}.$$ ...
1
vote
3answers
97 views

Finding derivative of given function.

f(t) = $\int_{t^2}^{4}\sqrt{\cos(x)+12}\;dx$ Rearranging limits of integration... $f(t) = -\int_{4}^{t^2}\sqrt{\cos(x)+12}\;dx$ Taking derivative... $f'(t) = -\sqrt{\cos(t^2)+12}\; - ...
0
votes
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.
1
vote
1answer
28 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 ...
0
votes
1answer
50 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?
0
votes
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 ...
0
votes
1answer
23 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
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 ...
0
votes
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 ...
11
votes
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 ...
0
votes
1answer
19 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
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 ...
1
vote
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 $$\phi'(a) = (1/g)'(a) = -\frac{g'(a)}{[g(a)]^2}.$$ ...
0
votes
2answers
423 views

Find the dimensions of a cylinder of given volume V if its surface area is a minimum.

The following is the question : Find the dimensions of a cylinder of given volume V if its surface area is a minimum. The cylinder has a closed top and bottom. 2 formula : (1) $V=r^2\pi ...
1
vote
1answer
40 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 ...
0
votes
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.
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 ...
0
votes
2answers
41 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
29 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 ...
3
votes
2answers
262 views

Complex differentiation under the integral sign (Ahlfors)

In Ahlfors' Complex Analysis text, page 202, he claims that in $\{ \Re z>0 \} $ $$\frac{d}{dz} \int_0^\infty \frac{2 \eta}{\eta^2+z^2} \frac{\mathrm d \eta}{e^{2 \pi \eta}-1}=- \int_0^\infty ...
4
votes
1answer
110 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 ...