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

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4
votes
3answers
75 views

$f:\mathbb R \to \mathbb R$ be twice differentiable , $f(x)+f''(x)=-xg(x)f'(x) , g(x) \ge 0 , \forall x \in \mathbb R$ , then $f$ is bounded?

Let $g:\mathbb R \to [ 0,\infty)$ be a function and $f:\mathbb R \to \mathbb R$ be a twice differentiable function such that $f(x)+f''(x)=-xg(x)f'(x) , \forall x \in \mathbb R$ , then is it true ...
3
votes
1answer
54 views

$f(x)$ is a differentiable function for $x\in[a,b]$, and $f'(a)=f'(b)$, prove: there is a $\theta$ such that…

$f(x)$ is a differentiable function for $x\in[a,b]$ ($f'(x)$ may not continuously), and $f'(a)=f'(b)$, prove: there is a $\theta$ such that $$f'(\theta)=\frac{f(a)-f(\theta)}{a-\theta}$$ I think we ...
1
vote
2answers
28 views

Finding all tangent lines that pass through a specific point (not the origin)

I was given the function $y = x^3-x$ and told to find all tangent lines that pass through $(-2,2)$. Not sure what steps to take past finding the derivative.
4
votes
1answer
72 views

Then is $f_a$ continious?

Excuse me for the bad title, here's the question Given a differentiable function defined on R. For a given number $a$, $\forall x\in \mathbb R, x\neq a$, by mean value theorem, there exists a ...
0
votes
3answers
40 views

Derivative of Equation

So i have this problem with the function: $$U(x)=\frac{A^2}{x^2} - \frac{A}{x}$$ I need to find the derivative of $U$ to find the min and max values. It says in the problem that $A$ is a positive ...
5
votes
6answers
108 views

Find the $n$th derivative of $f(x) =\frac{x^n}{1-x}$

Question Find the $n$th derivative of $f(x) =\frac{x^n}{1-x}$ What I've managed thus far First I thought that I might be able to discern a pattern by calculating the first few derivatives of ...
0
votes
0answers
43 views

Derivative norm(d^T*x)

Is the following derivative correct? $$ \nabla ||d^Tx||_2=\frac{1}{||d^Tx||_2}dd^Tx, $$ where $d \in R^n$ and $x \in R^n$. Thanks, Tom
0
votes
1answer
43 views

How to prove that $g' $ is not bounded on the interval $[-1,1]$?

Let $g:= R\rightarrow R$ be defined by $g(x):=x^2 sin (1/x^2)$ for $x$ is not equal to 0, and $g(0)=0$. Show that g is differentiable for all $x$ in $R$. Also show that the derivative g' is not ...
2
votes
5answers
50 views

differentiate *g(x)* if $g(x)=e^xf(e^{-x})$

differentiate g(x) if $g(x)=e^xf(e^{-x})$ Using any website to evaluate this derivative like wolframalpha.com we will get the result ===> $e^xf(e^{-x})-f'(e^{-x})$ But we know from the ...
2
votes
1answer
35 views

Finding a derivative of a series

I am trying to compute the derivative of the following series. Define: $$ u_n=\frac{1}{n^2}e^{-n^2x^2}\,\,x\in R $$ and $$u=\sum_{n=1}^\infty u_n$$ Then I am trying to compute $u'$. Any helps? ...
3
votes
2answers
104 views

Why is the derivative of a polar function $dy/dx$ and not $dr/d\theta$?

I don't understand. If $r = 2\cos(\theta)$ then why is the derivative: $dy/dx$? I have a "hypothesis," By the polar equation are you really describing a curve in the cartesian plane? So is that ...
1
vote
1answer
39 views

Derivative of Square Root Polynomial?

How do you find the derivative of $\sqrt{x^2 - 4x + 4}$ I applied Chain rule and got this $\frac{x-2}{\sqrt{(x-2)^2}}$ However, the fill-in box requires two distinct functions (piecewise) ...
1
vote
1answer
30 views

Another Fundamental Theorem of Calculus Proof

Let $f : R \to R$ be continuous and $\delta > 0$. Define $g(t)=\int_{t-\delta}^{t+\delta}f(x)dx$ for all $t \in R$. Prove that $g$ is differentiable and compute $g'$. I'm pretty sure you know that ...
0
votes
1answer
15 views

Implicit Derivative with Three Variable

Find $\frac{\partial z}{\partial x}$ from these equation: $xyz-2xz+3yz-4xy=0$ How $\frac{\partial z}{\partial x}$ can be done using manually method? Thanks... I stuck at variable whose contains ...
1
vote
2answers
366 views

Fundamental Theorem of Calculus Proof

Find $f'$ where is $f$ is defined on $[0, 1]$ as indicated: $$f(x) = \int_x^{\sqrt{x}} \frac 1{1+t^3}dt$$ I know that the fundamental theorem is going to be used in this proof, but I'm not really sure ...
0
votes
2answers
42 views

Derivative of a definite integral with two constraints

I am new to this, and I want to see if I have the right answer: $$\frac{d}{dx}\int^{2x}_{x} s^2 ds = \int^{0}_{x} + \int^{2x}_{0}= -\int^{x}_{0}+\int^{2x}_{0} =-x^2+8x^3 $$
0
votes
1answer
55 views

How to complete this proof

Let $n$ be in $N$ and let $f:R$ to $R$ be defined by $f(x):=x^n$ for $0\leq x$ and $f(x):=0$ for $x<0$. For which values of n is $f'$ continues at 0? For which values of n if $f'$ differentiable at ...
0
votes
1answer
36 views

An inequality for powers of reals

Let $a>b>0$ and let $n \in N$ satisfy $n \geq 2$. Prove that $$a^{1/n}-b^{1/n}<(a-b)^{1/n}.$$ If we let $a=x$ and $b=x-1$, then we need to show that $f(x):=x^{1/n}-(x-1)^{1/n}$ is ...
2
votes
1answer
30 views

Rate of change of rectangle inside triangle

A rectangle is inscribed inside a right angled triangle with hypotenuse 50cm and an angle of 30 degrees. I have supplied a diagram below. The vertical line marked h is moving to the right at 3cm per ...
1
vote
2answers
96 views

Why are integral and differential operators commutative?

For instance, let's assume a constant 3D surface over time $S$. $$ \frac{d}{dt}\iint_S \mathbf B \cdot \mathbf{ds} \quad=\quad \iint_S\frac{\partial \mathbf B}{\partial t}\cdot \mathbf{ds} $$ Why ...
0
votes
2answers
72 views

An odd function $f$ is differentiable at zero. Prove $f'(0)=0$?

I know that $f'$ of an even function is odd function, thus I have $f(x)=f(-x)$. However I'd no idea how to prove that $f'(0)=0$? Please answer my question...
0
votes
2answers
36 views

Chain rule using the expression F=150W^1/3

Suppose the attendence of a baseball game was denoted by W alone in the format F=(150W)^1/3. Is this function (strictly) concave or convex. Explain. To which I answered that it would be strictly ...
0
votes
1answer
51 views

Expectation of the derivative of a random process

Let's have a Random Process $Y(t) = X(t) + 0.3 X'(t)$ Mean of $X(t) = 5t$ Question : Find the mean function of $Y(t)$ What I did : $E(Y) = E(X) + 0.3\cdot E(X')$ ? I don't know if I have ...
2
votes
2answers
62 views

Why is Implicit Differentiation needed for Derivative of y = arcsin (2x+1)?

my function is: $y = arcsin (2x+1)$ and I want to find its derivative. My approach was to apply the chain rule: ${y}' = \frac{dg}{du} \frac{du}{dx}$ with $g = arcsin(u)$ and $u = 2x+1$. ${g}' = ...
0
votes
0answers
32 views

Continuity of Derivatives

I am going over a statement in Rudin which says "Suppose $f$ is a real differentiable function on $[a,b]$ and suppose $f^{'}(a)<k<f^{'}(b)$. Then there is a point $x\in (a,b)$ such that ...
-1
votes
2answers
45 views

How to show that $f(x) = x|x|$ is differentiable at 0?

So I've gotten $$f'(x)=\dfrac{2x^2}{|x|}$$ How to show that the following function is differentiable at 0?
0
votes
3answers
68 views

Differentiability of a function on $\mathbb R$ such that $f(x+1)=f(x)$.

Let, $f:\mathbb R \to \mathbb R$ be a function such that $f(x+1)=f(x)$ for all $x\in \mathbb R$. Then which of the followings are correct? (a) $f$ is bounded. (b) $f$ is bounded if it is continuous. ...
0
votes
1answer
42 views

Show that this function is infinitely often differentiable

I have the function $g(r):=e^{-\frac{1}{1-r}}$ for $r \in [0,1)$ and $0$ for $r \ge 1$. Now, I want to show that $g \in C^{\infty}([0,\infty))$. I guess this can be shown somehow by finding an ...
2
votes
2answers
20 views

First derivative of a multivariable function

This question was featured on my calculus mid-term today : Find the first derivative of $g(x,y)$ where : $$ g(x,y) = f(x^2 + y^2 ,xy) $$ This is the exact text of the problem. I just don't ...
0
votes
1answer
22 views

Necessary condition of optimality for functionals

Let $C(a, b)$ denote the set of all surjective and continuously differentiable functions $\alpha:[a, b] \rightarrow [a, b]$. Consider the functional on $C(a, b)$ $$ F[\alpha(t)] = \int_a^b ...
1
vote
1answer
41 views

Prove $f''(x)>0$ for some x.

Question: Let $f$ be a twice differentiable function on $R$. Suppose $f(0) = 0$ and $f(x)/x$ is increasing for $x > 0$, prove $f′′(x) ≥ 0$ for some $x > 0$, but not necessarily for all $x > ...
3
votes
2answers
150 views

Looking for differentiable functions $f$ such that the set of points at which $|f|$ is not differentiable has some particular properties

I know that $\sin x$ is differentiable at all points but $|\sin x|$ is not differentiable at countably many points namely at integer multiples of $\pi$ . So I am asking the following questions i) ...
0
votes
1answer
31 views

Finding $f(x)$ from first and second derivitive

What is f(x) when $$f(1)=0$$ first derivative $$f(1)=1/2$$ second derivative $$f(x)=1/x^3$$ Currently i have tried where the second derivitive = first derivitive + constant at x=1 1/x^3 = f^i(x) + ...
1
vote
2answers
58 views

How the derivative might fail to exist

Can a function have both a vertical tangent and cusp? Does The function $3x^{1/3}(x+2)$ have a vertical tangent and if so why? I believe that it has a cusp.
2
votes
1answer
24 views

Computing derivative of parametric equation

This is probably a silly question but I am just not sure if I understand what to do. So I have the parametric equations: $x=6\cos (t)-2\\ y=5\sin (t)+3$ I am asked to compute $\dfrac{dy}{dx}$ at ...
5
votes
1answer
45 views

Using the Inverse Function Theorem prove that $(\sin^{-1}x)'$ = $\frac{1}{\sqrt{1-x^2}}$.

Using the Inverse Function Theorem prove that $(\sin^{-1}x)'$ = $\frac{1}{\sqrt{1-x^2}}$. Proof: Let $f(x) = \sin x$, for $x$ in $(-1,1)$. Then let $x_{0}$ be in (-1,1). Then $f'(x_{0})$ = ...
3
votes
1answer
32 views

Derivative of ellipse

Find $y'=\frac{dy}{dx}$ when: $$2x^2-xy+y^2=1$$ How do I solve this? Do I start with this?: $$y=\frac{2x^2+y^2-1}{x}$$ $$y'=\left(\frac{2x^2+y^2-1}{x}\right)'$$
2
votes
3answers
47 views

addition of two differential functions is differentiable

I am stuck with proving the following statement. In fact, I am proving another assumption, and the proof of this would help me to proceed. Assume that $f_1$ and $f_2$ are differentiable on the ...
2
votes
1answer
42 views

Finding a tangent of an algebraic curve: $xy = 1$ [Well written explanation]

I want to find, using (easy) calculus, the slope of a tangent to the algebraic curve $xy = 1$. The tangent I want to find is the tangent that passes through the point $(x_i,y_i) = (0,f(t))$. $f$ is ...
5
votes
3answers
112 views

Prove that $\frac{d^n}{dx^n} \cos(ax) = a^n \cos (ax + \frac{n\pi}{2})$

Question Prove that $\frac{d^n}{dx^n} \cos(ax) = a^n \cos (ax + \frac{n\pi}{2})$ I could calculate the first few derivatives of $\cos(ax)$ and consequently observe the pattern that unfolds, ...
0
votes
1answer
40 views

A question about the elementary symmetric polynomial

I have asked this question and have come up with a possible answer $$ \frac{d^j}{dx^j}[\frac{(x)_c}{j!}] = e_{c-j}(x,x-1, \cdots ,x-c+1) $$ My first question is, how can I prove this? It seems trivial ...
0
votes
0answers
21 views

Finding the value of the inverse function with inverse function theorem

I am stuck by the following problem. Let $h:\Bbb R^2\rightarrow \Bbb R^2$ and $$h(x,y)= (x^2+3xy+xy^3, x^3-5y^2)$$ Let $g=h^{-1}$ near $(0,1)$. Find $Dg(0,-5)$ I showed that the inverse function ...
1
vote
3answers
35 views

Differntiable and continuous

Is it true that a function which is not continuous at a point will not be differentiable at that point? Graphically it seems so, but can we prove this formally? Also, if the above statement is ...
1
vote
1answer
24 views

Derivative relation between two equal functions

I am stuck with the following problem. Suppose $g: \Bbb R\rightarrow\Bbb R$ is $C^1$. $f(x,y)=g(x^2+y^2)$. I need to show that $xf_y=yf_x$ My attempt was: $f_x=g_x \cdot 2x$ (1) and $f_y=g_y\cdot ...
0
votes
1answer
39 views

0th-order differential operator vs. 1st-order differential operator on a vector bundle $(E, \pi,M)$

Consider a vector bundle $(E,\pi,M)$. A 0th-order differential operator on $E$ is a $C^\infty(M)$-linear endomorphism $\Gamma E\rightarrow\Gamma E$. $\Gamma E$ is the set of sections on $M$. A ...
0
votes
1answer
23 views

Differentiability of two variable function with two possibilities

There is a another question which is exactly similar to my question in this website, but I think I am still confused about that too, I couldn't get it. I would be very very very thankful if someone ...
1
vote
3answers
34 views

Equation of line normal to $y = x^3 -2x^2$ at $x=0$

Find the equation of normal to the curve $y =x^3 - 2x^2$ at $x= 0.$ Find the co-ordinates of the point of intersection of the normal and the line $y = 4.$ I differentiated the equation with respect ...
0
votes
2answers
40 views

Studying the differentiability of a function at a point $(a_{1},a_{2})$

I have a function $\ f: \mathbb{R}^2 \to \mathbb{R} $ to study: 1) It's continuity at the point $(a_1,a_2)$. 2) The partial derivative exists at $(a_1,a_2)$? 3) Are the partial derivatives ...
1
vote
0answers
22 views

Where do the minus sign and Laplacian come from in this derivative?

I want to find this functional derivative: $$\dfrac{\delta \int d^d x'[\nabla_{x'} \phi(\vec{x}')]^2}{\delta \phi(\vec{x})} = \int d^d x' \left(\dfrac{\delta \nabla_{x'} \phi(\vec{x}')}{\delta ...
0
votes
1answer
25 views

Minimizing an open box (Calc I)

A rectangular container with an open top is to have a volume of $12 \;\text{m}^3$. The length of its base is twice the width. Material for the base costs (in dollars) 10/$\text{m}^2$. Material for ...