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

learn more… | top users | synonyms (2)

0
votes
1answer
55 views

Show that the approximation to $f$'($x_0$) has discretization error $O$($h^2$)

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h$ and $x_{1} = x_0 + h$ with $h$ > 0. Given a smooth function f, show that the approximation to $f'(x_0)$ given by the centered ...
2
votes
1answer
61 views

How to find $g'(0)$ if $g(x)=\sin(f(x^2+x)-2x)$ and $f$ satisfies $|f(x)|\le x^2$ for all $x$?

The problem goes as follows: Let $f(x)$ be a function such that $|{f(x)}|\le x^2$ $\forall x \in [-1,1/7]$. The first part of the problem is prove that $\lim_{x\rightarrow0} ...
1
vote
1answer
34 views

How to normalise equations of the form $dy/dx=B$ and $d^2y/dx^2=A$?

So I am trying to normalise equations of the form, $$dy/dx=B \mbox{ and } d^{2}y/dx^{2}=A$$ If I define $y^{*}$ as; $$y^{*}=By \Rightarrow dy^{*}/dy=B $$ Is it also then true that, $$d(dy^{*})/dy = B ...
2
votes
2answers
95 views

Direction of Greatest Increase

Problem: Find the direction of greatest increase at $P$. $$f(x,y)=4x^2+y^2+2y$$ $$P=(1,2,12)$$ Solution: The greatest increase in $f(x,y)$ at $P$ can be attained by moving in the direction of ...
6
votes
4answers
253 views

How to maximize or minimize $f(x)=ax^2+bx$?

I am trying to self-study calculus from the Internet. I have learnt things mostly from MIT OCW site and also from other sites. However, I am stuck on this simple problem: Find the maximum/minimum ...
3
votes
3answers
96 views

Using the chain rule with a composite function

I'm a little confused on this homework problem and I could use some explanation if anyone has seen something like it before. The question is: Use the Chain Rule to find $\frac{dy}{dt}$ at $t = 9$ ...
0
votes
3answers
48 views

Find the derivative of $\frac{{(x^3)^{4/3}}}{(2-x)^{4/3}}$

I tried to solve it using the chain rule first and then doing the quotient rule after. However, I end up with $\frac{24x^2-8x^3(x^3)^{4/3}}{3(2-x)^{7/3}}$ My professor said it's wrong. Kindly explain ...
3
votes
1answer
69 views

Derivative: $f_x, f_y, f_{xy}$ of function - $f(x,y)$

Let's say $f(x,y) = x^2 + 2xy +y^2$ $f'_x = 2x + 2y$ $f'_y = 2y + 2x$ $f'_{xy} = 2x + 2y$ ? Am I right about the third?
4
votes
0answers
36 views

Distributions and primitives

I was wondering: if distributions are seen as a generalization of functions that "removes obstructions" to the operation of derivation, is there a generalization of functions that would remove any ...
1
vote
3answers
97 views

Evaluate $\frac{\partial^2}{\partial t^2} \left[ \prod_{j=1}^k (1+t+\dots+t^{d_j -1}) \right]$ at $t=1$

I need to find a "nice" formula for the evaluation of $\frac{\partial^2}{\partial t^2} \left[ \prod_{j=1}^k (1+t+\dots+t^{d_j -1}) \right]$ at t=1, where $d_j \in \mathbb{N}$. I have already proved ...
0
votes
2answers
326 views

Optimization problems: Finding the optimal path

I'm still trying to get the hang of optimization problems in calculus and I'm looking for a little help. I'm having trouble finding equations to model the following problem: I'm fairly sure I need to ...
1
vote
1answer
71 views

Proving power rule for $x^n$ with arbitrary positive $n>0$

How to prove that $\frac{d}{dx} x^n = nx^{n-1}$ for every $n>0$ (possibly fractional)? Context It was already shown that $\frac{d}{dx} x^n = nx^{n-1}$ for positive integer $n$. My friend ...
1
vote
0answers
53 views

Differentiation and integration of a series

If I have a power series $$\sum _{k}^{\infty }f(x)$$ and I differentiate it I get according to my current knowledge $\sum _{k}^{\infty }f(x)'$,however when I look at a power series defined by $$\sum ...
5
votes
1answer
446 views

Combination of linear functions that give the derivative operator

Let $D$ be the derivative operator and $C^\infty$ the set of functions derivable once. Here $f^n=f\circ f\circ\cdots\circ f\text{, }n\text{ times}$ It can be easily shown that there exists ...
0
votes
2answers
103 views

Ordinary Chain Rule Confusion

Let $f$ be the function defined in Q1, and let $g$ be a function such that $g^\prime(x)=\sin(\sin(x+1))$ and $g(0)=2$. Find $(f\circ g)^\prime(0)$ and $(g\circ f)^\prime(0)$. For $x\neq0$, the ...
0
votes
3answers
155 views

Assumptions in Word Problems.

My dilemma has been that I am confused on how we make mathematical assumptions in WORD problems. Suppose you are given a related-rates word problem. (Q#) Air is being pumped into a spherical balloon ...
2
votes
4answers
74 views

Prove that $f'(x_o) =0$

Let $f$ be a function defined on an interval $I$ differentiable at a point $x_o$ in the interior of $I$. Prove that if $\exists a>0$ $ \ [x_o -a, x_o+a] \subset I$ and $ \ \forall x \in [x_o ...
0
votes
1answer
821 views

Rate of change of area of a square with respect to side length

I have been asked to find the rate of change of the area of a square with respect to the length of its side when the side is 4ft. This is how I thought I should do it. Area=$s^2$ ...
0
votes
2answers
65 views

Prove that $\lim_{x \to \infty} f(x) = \infty $

Let $f: \mathbb R \to \mathbb R$ be a function such that there exists constants $ b,\ m> 0,\ c \in \mathbb R $ such that $$f(x) > mx+c,\ \forall x>b$$ Prove that $\lim_{x \to \infty} f(x) ...
1
vote
1answer
65 views

Domain of densely-defined second derivative operator, and its factorization

Let $$-d_x^2: \{f \in L^2[0,1];f \in AC^1[0,1] , f(0)=f(1)\} \rightarrow L^2[0,1]$$ be the second derivative operator. Here $AC^1[0,1]$ is the space of functions whose first derivative is absolutely ...
0
votes
0answers
37 views

Covering up discontinuities to create analyticity

The floor function, $\lfloor x \rfloor$ , has a "jump" at the integers where its derivative ceases to exist. Everywhere else, its derivative is zero. Now, I wish to multiply the floor function by ...
1
vote
2answers
103 views

How to prove a function is not differentiable

Given: $$f(x) = \begin{cases} x^3 &&& x<2 \\x+6 && &x \geq 2 \end{cases} $$ I need to prove that $f(x)$ is not differentiable at $x=2$, what should I do? $$\lim_{x \to ...
0
votes
1answer
70 views

how to differentiate an indicator function?

I'm reading this paper and I arrived at this part when they introduce a formula for what they call 'an indicator function'. Here is a shot: what I understood from the first two formulas is that I ...
1
vote
4answers
62 views

Question about tangent and slope

Given the graph of $y=-e^{-x}$ and that there is a tangent to the graph that crosses the x axis at $(-4,0)$ determine the slope of that line. So this seems like a simple question but I don't know why ...
0
votes
0answers
23 views

Linearizing non-linear least squares: Problem with derivatives

We want to approximate $$y_i \approx a b^{x_i}$$ and thus have $$S=\sum_{i=1}^m (ab^{x_i}-y_i)^2$$ as least squares error term. This term is not linear in b, so it is not easy to calculate its ...
2
votes
2answers
90 views

A function is real-differentiable iff it has a complex-differentiable extension

Is this conjecture true? A function $f:\Bbb R\to\Bbb R$ is real differentiable at $a$ if and only if there exists a complex-differentiable function $g:A\to\Bbb C$ for some neighborhood of $a\in ...
1
vote
1answer
44 views

Question about the Fundamental Theorem of Calculus

So I have studied the FOTC, but not really sure of what I read so this question is just to help me learn the FOTC and understand how to do problems like it. $$ if $$ $$F(x)=\int_0^x\sqrt{sin^3(t)}dt$$ ...
4
votes
2answers
107 views

How can you explain implicit differentiation?

So I am taking calculus 1 online from a local college (bad idea, but the only thing that fit my schedule). The professor used the notation $f'(x) =$ for EVERY function up until two weeks ago. All of ...
1
vote
2answers
48 views

Is there a continuous compact supported function $f: \mathbb{R}^n\rightarrow \mathbb{R}^{2n}$ such that $f^{-1}$ is continuous differentiable

Is there a continuous compact supported function $f: \mathbb{R}^n\rightarrow \mathbb{R}^{2n}$ such that $f^{-1}$ is continuous differentiable? I don't know which theorem is related to this question, ...
0
votes
1answer
39 views

Can you uniquely define a tangent line at a point for a 3D csurve?

Let f be a function of the form: $x=f_x(t); y=f_y(t);\text{ and }z=f_z(t)$. Does the derivative set of the 3 functions mean the tangent at a point on the curve of f? Thank you in advance.
4
votes
1answer
111 views

Why generalize the derivative for multivariable functions? [duplicate]

Sorry if this is a dupe (did a search, couldn't find anything). In single variable calculus, if the following limit exists: $$\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h},$$ then this expression ...
-1
votes
1answer
107 views

How to find the derivative of the function $ \int_{x}^{x^2}\frac{t}{\ln(t)}dt$?

The problem is to find $\displaystyle\frac{d}{dx}\int_{x}^{x^2}\frac{t}{\ln(t)}\,dt$ I could do this if I had the first clue on how to integrate $\dfrac{t}{\ln(t)}$ but even wolframalpha is giving ...
14
votes
2answers
441 views

Prove ${\large\int}_0^1\frac{\ln(1+8x)}{x^{2/3}\,(1-x)^{2/3}\,(1+8x)^{1/3}}dx=\frac{\ln3}{\pi\sqrt3}\Gamma^3\!\left(\tfrac13\right)$

Here is one more numerically discovered conjecture that I was not able to prove, and asking you for help: ...
0
votes
3answers
69 views

Finding the tangent line to the graph of $f(x)=(x+2)^{3/5}$ at $x=-2$

Does the graph of the function $f$ have tangent line at the given points? If yes, what is the tangent line? $f(x)=(x+2)^{3/5}$ at $x=-2$ solution: yes, $x=-2$ The derivative I found: ...
3
votes
7answers
123 views

Find the derivative of $y=x\sqrt{9-x}$

"Find the derivative of $y=x\sqrt{9-x}$." So this is what I have and now I'm stuck. \begin{align} y' &= x \frac{d}{dx}\left[(9-x)^{1/2}\right] + (9-x)^{1/2} \frac{d}{dx}(x)\\ &= x ...
3
votes
2answers
120 views

Show that the graph of $y=x^3\sin(\pi/x)$ extends to a smooth arc

Here's the problem: Let $y(x)$ be a real-valued function defined on the interval $x\in [0,1]$ by means of the equation $$y(x)= \left\{ \begin{array}{lr} x^3\sin(\frac{\pi}{x}) ...
1
vote
1answer
60 views

Evaluate Derivative Using $\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ Definition

Evaluate the derivative of $x^3 - 3x +1$ using the $\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ definition to find the tangent of the curve at the point $(2, 3)$. I already calculated this derivative ...
0
votes
1answer
50 views

Left & Right Area Approximation Using Y-Axis - Method Alternatives

Is there a simpler way of solving this then calculating x1(h)+x2(h)+x3(h)+x4(h) by using the given y values (in this case h, the height is one, because the length of each rectangle is one) ...
1
vote
2answers
99 views

Derivative of the trace of $X^TP^TPX$ with respect to P

$\newcommand{\Tr}{\operatorname{Tr}}$ Consider the following expression: $\Tr(X^TP^TPX)$ where $X$ and $P$ are real matrices. What is the best way to approach the calculation of its derivative ...
0
votes
1answer
1k views

Related Rates of Change - Cylinder Question

A cylindrical tank with radius 5 cm is being filled with water at rate of 3 cm^3 per min. how fast is the height of the water increasing? I dont want this question solved, but please help me correct ...
1
vote
1answer
51 views

Differentiable functions and examples

can someone give me an example of Differentiable function at x=4 and funcstions who dont Differentiable function at x=4? $f(x) = 2x-7$ $k(x) = 100x^7-55x^5+10000x^2$ $g(x) = 23$ Those are ...
0
votes
2answers
38 views

derivative of this special function

I would like to take the first derivative of the following function respect to x. what is the derivative of this function with respect to ...
1
vote
1answer
91 views

L'Hopital's Rule with $\lim \limits_{x \to \infty}\frac{2^x}{e^\left(x^2\right)}$

(a) Show that $$\lim \limits_{x \to \infty}\frac{2^x}{e^\left(x^2\right)}$$ is a standard indeterminate form, but that L'Hopital's Rule does not give you any information about the limit. (b) Show ...
0
votes
3answers
107 views

simplify the expression $\arctan\frac{x\sin t}{1-x\cos t}$

Same as above, how to simplify it. I am to calculate its $n$th derivative w.r.t x where t is const, but I can't simplify it. Any help would be appreciated. Thank you.
3
votes
1answer
69 views

What's the Differential of this Map $f:S^3\rightarrow \mathbb{R}$

$f:S^3 \rightarrow \mathbb{R}$ is defined as $f(x,y,z,w)=x+zw$, where $S^3= \{(x,y,z,w) | x^2 +y^2 +z^2 +w^2 =1\}$ I tried using a stereographic chart but that got ugly. The function is so simple I ...
2
votes
2answers
131 views

Differentiability of the sum of the series $\sum_k \sin(kx)/k^2$

How to show the following: If $ f(x) = \displaystyle\sum_{k=1}^{\infty} \dfrac {\sin(kx)}{k^2} $, then show that $f(x)$ is differentiable on $(0,1)$ I guess it should be related to uniform ...
0
votes
3answers
74 views

Find the derivative of $\frac{x^{1/3}} {({x^3+1})^{1/3}}$

I tried to solve it my answer is $$\frac{-2x^{4/3}(x^{3}+1)^{2/3}+1}{3x(x^3+1)^2}$$ I just want to make sure if I derived it correctly thanks
1
vote
0answers
52 views

Minimize distance between the ships

I'm studying calculus from a the book "Calculus with Analytic Geometry by Georfe F Simmons", and I have a certain difficulty to solve the following problem: Ah noon the ship A is at a distance at ...
1
vote
2answers
102 views

Derivative of matrix product: is it true that $\frac{d}{dt}(A^TA) = 2A^T \frac{dA}{dt}$?

$A$ is a square matrix. All elements of $A$ depend on a parameter $t$, that is, $a_{ij}=a_{ij}(t)$. Let $S(A):=A^TA$, and take the derivative of $S$ w.r.t. $t$: $\displaystyle \frac{dS}{dt}$ Now, ...
1
vote
2answers
115 views

Finding the differential of $y=(u+1)/(u-1)$

I'm having trouble with differentials. I've been trying to learn about them online using great resources like PatrickJMT but I'm having trouble finding examples for this kind or problem. I hate asking ...