Tagged Questions

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

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0
votes
3answers
42 views

Is this method for computing limits valid?

I was shown this 'proof' for proving the limit $\lim_{x \rightarrow 1}\dfrac{\ln x}{x-1} = 1$ "The expression on the left is the statement of differentiation by first principles for the gradient of ...
-1
votes
1answer
22 views

Determine if a function is differentiable

If $$\lim_{\delta x^+\to 0}\frac{f(x+\delta x)-f(x)}{\delta x}=\lim_{\delta x^-\to 0}\frac{f(x+\delta x)-f(x)}{\delta x}$$ at the point $a$ is this a necessary and sufficient condition for $f(x)$ ...
0
votes
1answer
13 views

Sequence of Polynomials and Weierstrass Approximation Theorem

Using the Weierstrass Approximation Theorem I need to prove that if $f$ has $k$ continuous derivatives in $[a,b]$, then exists a sequence of polynomials {$P_n$} such that {$P_n$} converges uniformly ...
0
votes
3answers
35 views

How to simplify this derivative?

I don't understand how I can go from the first step to the step on the right, can someone help me please? $$ \frac{dy}{dx} = \frac{3 \sqrt{x}}{2\sqrt{1+x^2}} - \frac{\sqrt{x^5}}{\sqrt{(1+x^2)^3}} = ...
6
votes
1answer
46 views

A function is smooth at a point and not smooth in any neighbourhood of it, exist or not?

Suppose that a function $f$ defined in an open set $U \subseteq \mathbb{R}^m$ is smooth at a point $p \in U$. Then we have that there exists an open set $U_n \subseteq U$ $($ say $U_{n+1} \subseteq ...
3
votes
1answer
28 views

Find all numbers $c$ that satisfy Mean Value Theorem

Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers $c$ that satisfy the conclusion of the Mean Value Theorem. My function is a ...
1
vote
1answer
19 views

Finding maximum values of a function on an interval?

I took the derivative of the original function and then set it to zero and got $$r= -2r0/((dv/dr0)-2)$$ Honestly, I'm pretty lost and need some help getting started. Thank you.
0
votes
1answer
31 views

A problem relating to mean value theorem

Suppose that $f(x)$ is continuous in $[0,1]$ and differentiable in $(0,1)$, and $f(0)=0$, $f^{'}(x)>0 (x\in(0,1))$. Show that there exists $\xi,\eta\in(0,1)$, $\xi+\eta=1$, such that $$ ...
-2
votes
0answers
25 views

For a function how to find concavity and points of inflection [on hold]

For the function $$f(x)=\frac{x^2}{x^2+3}$$ Find the intervals on which $f(x)$ is increasing or decreasing. Find the points of local maximum and minimum of $f(x)$. Find the intervals of concavity ...
0
votes
0answers
9 views

Derivative of Trigonometric Functions, Increasing and Decreasing Functions

Plants are photoautotrophic which means they produce their own food using light. Photosynthesis is the mechanism by which plants produce food using water, a carbon source, and sunlight. In London, ...
1
vote
4answers
31 views

Find the derivative of $y = x^{\ln(x)}\sec(x)^{3x}$

What is the derivative of $$y = x^{\ln(x)}\sec(x)^{3x}$$ I tried to find the derivative of this function but somewhere along the way I seem to have gotten lost. I started off with using the ...
0
votes
0answers
23 views

$d/dx((\sinh^{-1}(\tan x)))$

I'm just wondering if I did everything correctly for this question, I know the answer is correct but I'm not 100% sure the steps I took to get there are valid: \begin{align} d/dx(\sinh^{-1}(tanx)) ...
0
votes
1answer
7 views

Finding local Maxima and minima then create a table

Use the First Derivative Test to find the points of local maxima and minima of the function $ƒ(x)=4x^3+3x^2−6x+1$. The final answer is expected to be in the form of a table containing all the ...
2
votes
0answers
11 views

Product rule for Hessian matrix

Let $f: \mathbb{R}^n \to \mathbb{R}$ and $g: \mathbb{R}^n \to \mathbb{R}$. Is there a general formula for the Hessian matrix of their product? That is, what is $H(f(x) g(x))$, where $H(f(x)) = ...
9
votes
3answers
56 views

Why does $n$-time differentiation of product have the same structure as raising sum to $n$th power?

A formula for differentiating a product is well known: $$(ab)'=a'b+ab'.$$ At first sight it doesn't resemble anything interesting. But what if we differentiate twice? We'll get ...
1
vote
0answers
26 views

Differential operators

I have a few very simple questions I think. First and foremost what exactly is $|D^2u|^2$? I know $D^2u$ is the Hessian matrix. Secondly, how do I show that $\sum_{i=1}^n u_{x_i}\cdot Du \cdot ...
3
votes
2answers
39 views

Calculate the derivative and find its domain: $\;f(x)= \sqrt{\ln(x)+2}$

I calculated the derivative as $$f'(x) = \frac{1}{2x \sqrt{2+\ln x}}$$ How do I find out the domain?
0
votes
5answers
674 views

why there is no derivative in sharp turns?

why there is no derivative in sharp turns in functions? I understand that it may be difficult or impossible to actually draw a tangent at that point, but is there a mathematical proof that there is no ...
0
votes
1answer
12 views

Derivative of Lebesgue integral function at the endpoints

Let $f$ be a non-decreasing Lebesgue-integrable real function on $[a,b]$. I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа (p. 340 here) that $$\lim_{h\to ...
0
votes
1answer
22 views

Is the square root of the absolute functiom differentiable at x = 0?

I've been trying to solve the problem below for hours but so far I haven't managed to find a solution. Help would really be appreciated. Thanks a lot! Problem Show whether or not the function ...
0
votes
1answer
16 views

Derivatives using the chain rule

$F(x) = 16.5x^2$ $x(Y) = 0.065Y + 0.68$ find $dF/dY$ derivative of $F(x) = 31x$, derivative of $x(Y) = 0.065$ I don't understand how to find $dF/dY$.
0
votes
1answer
9 views

Related Rates Problem about moving shadow

I have another question about related rates. I have been asked the following question about related rates. It's been a while since I looked at related rates. I appreciate if anyone can help me with ...
0
votes
1answer
63 views

Differentiation of a circle

As a discus thrower is spinning counterclockwise to throw a discus, the discus travels along the path given by the circle $x^2+y^2=4$. If the discus is released at the point $(\sqrt2,\sqrt2)$ and ...
0
votes
2answers
32 views

Related Rates Problem involving two runners on a circular path

Problem: There was a typo in the original statement. I fixed it now!! Two runners start running (from the same point) in opposite directions along a circular path of radius $100\ m$ at a speed of ...
0
votes
2answers
40 views

Optimizing a box

I'm learning the use of derivatives and I have found a problem: Supposing we want to build a box of $4000\, \textrm{cm}^3$ of volume without top and a square base. Which are the measures so we ...
2
votes
1answer
37 views

Nowhere differentiability of Weierstrass function

It's again from Tao's book. 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 interval ...
0
votes
2answers
36 views

How do I show that $f'(x)=f(x)f'(0)$ from $f(x+y)=f(x)f(y)$ [on hold]

I already know $f(0)=1$, but let's assume that $f'(0)=k$, how do I get to $f'(x)=kf(x)$?
1
vote
1answer
43 views

What's the derivative of $f(x) = \cos (x) ^{\ln(3x)}$? [on hold]

I don't understand where or how many times I need to apply the chain rule.
1
vote
1answer
15 views

How to find extreme values and where they occur?

What are the extreme values of the function $$y = (x^2 - 1)^{1/2} = \sqrt{x^2 - 1}$$ and where do they occur? I have gotten as far as finding that $$\frac{dy}{dx} = \frac{x}{\sqrt{x^2 - 1}}$$ What ...
1
vote
1answer
32 views

Differential equation $x\cdot f'(x)\cdot\left(f(x)+1\right)=f(x)$

In a proof of the series expansion of the Lambert-W-function, I need that it is the only non-zero function satisfying: $$ x\cdot f'(x)\cdot\left(f(x)+1\right)=f(x) $$ Is it true?
0
votes
2answers
33 views

Derivative of x^(2x)

Maybe a simple question but I can't find a good example of it on the internet. What is the derivative of: $x^{2x}$ What is the simplest method of determining the derivative and what kind of rules ...
1
vote
0answers
17 views

Verify the following rate question?

Can someone help point out where I went wrong? Question: Sand is being poured from a conveyor belt at the rate of 100 cubic feet per minute. The sand form a pile that is the shape of a cone whose ...
0
votes
2answers
51 views

Differentiate $y =\sin(1+x^2)^{1/2}$

I've tried differentiating $y= \sin(1+x^2)^{1/2}$ using the chain rule, but I keep getting the wrong answer. Can anyone give me a step by step so I can see what I'm doing wrong? Thanks.
1
vote
1answer
28 views

Limits in Differential calculus

I have to prove that $$ f'(x)=\lim_{h\rightarrow 0}\frac{f(x)-f(x-h)}{h} $$ At first I used the standard definiton $$ f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} $$ Then I replace $-h=u$, ...
3
votes
4answers
45 views

Differentiate y=Cot²(sinx)

$$ y = \cot^2(\sin x) $$ How do I differentiate that? I tried using chain rule but I don't understand how to differentiate $\cot^2(\sin x)$.
0
votes
1answer
47 views

Differentiation using first principal [on hold]

Find the derivative of the following function $$f(x)=x+\sqrt{x}$$ using first principle (Limit of difference quotient).
0
votes
2answers
17 views

Solution of $y'=xy^{1/3}, y(0)=0$ equal to $0$ in $[-c,c]$ and positive for $|x|>c$.

I'm looking for a continuous function $y(x)$ which satisfies the above and trying to make it depend on $c$ so that a solution exists for any $c>0$. I read it is possible, but I can't do it... Can ...
3
votes
1answer
23 views

What is the definition of a cusp?

So there are 3 situations that a function is not differentiable 1. a vertical tangent, 2. a discontinuity and 3. at a cusp. Consider the function $f(x)=e^{-|x|}$ at the point $x=0$. It is continuous ...
-5
votes
1answer
25 views

What is the original function of the derivative 5x? [on hold]

The derivative is 5x but what is the function?
1
vote
1answer
19 views

Minimization of the difference between two functions

I am stuck with the following problem. I need to find the point on the line passing through $(1,0,0)$ and $(0,1,0)$ that is closest to the line through $(0,0,0)$ and $(1,1,1)$ Please do not give full ...
0
votes
0answers
15 views

finding max and min value of polynomial function with starting point and steps

My instructor said you will find max and min value of a polynomial function. However, you will use a point that is called starting of search and step size. I don't understand starting of search and ...
3
votes
1answer
28 views

Calculating derivative by keeping all but one $x$ constant and adding the results together for different $x$

In this answer there's a comment which says That's not wrong; that's a perfectly valid method. You get the derivative of any expression with respect to $x$ as the sums of all the derivatives with ...
0
votes
2answers
28 views

Derivatives using chain rule. $f(x) = (7x^3 + 2)^3(6x^2 - 1)^4$

I understand how to use the chain rule with simple functions. But in this there are two core's too choose from. Can't wrap my head around how to even start. Please help, been stuck on this for too ...
-1
votes
2answers
44 views

How to solve for $e^{-x}$?

I'm sure I learned this at some point, I think it has something to do with $\log$-ing the function. Could someone please walk me through it? $$0=4x^2e^{-x^2}-2e^{-x^2}$$ By the way, this is the ...
0
votes
1answer
29 views

Derivative of $F : \mathbb R^{n+m} \rightarrow \mathbb R^{n+m}$

Question : Let f be continously differentiable function on an open set E $\subset \mathbb R^{n+m}$ into $\mathbb R^n$. Define a funtion F on E into $\mathbb R^{n+m}$ such that F(x,y) = ...
0
votes
2answers
17 views

Finding the derivative of the equation.

I know the answer, but I don't know how to get to it. If someone could explain it to me would be extremely helpful. Thanks! $y=(\arctan(x))^3$
0
votes
4answers
50 views

Showing $\frac{d^6y}{dx^6}$ using $\frac{d^4y}{dx^4}$ and $\frac{d^2y}{dx^2}$

I found : $$\frac{d^2y}{dx^2} = -x \sin x + 2 \cos x$$ $$\frac{d^4y}{dx^4}=x \sin x - 4 \cos x$$ I have seen the pattern, but how do I use these results to find $$\frac{d^6y}{dx^6}$$ ...
1
vote
2answers
17 views

Using the limit definition to find the derivative of this function.

f(x)=sin3x you have to use the limit definition to find it and I plugged x+h in and everything and I used the sin sum formula but I just can't get rid of the h on the bottom!
0
votes
1answer
14 views

Find the area of the triangle formed

Find the area of the triangle formed by the x-axis, the y-axis, and the tangent line to the graph of $y = \frac{1}{\sqrt{x}}$ at the point where $x=k$. I have no clue how to do this and my teacher ...
3
votes
2answers
19 views

Equations of the tangents to the curve $f(x) = 2x^2 + 3$

Find the equations of the tangents to the curve $f(x) = 2x^2 + 3$ that pass through the point $(2,3)$. Include a sketch as part of your solution. First, I found the derivative of $f(x)$ which is ...