0
votes
2answers
72 views

$f(x)$ is everywhere differentiable on $[a,b]$ then give examples

$f(x)$ is everywhere differentiable on $[a,b]$ then give examples for each (they are independent) (1) $f'(x)$ is not Riemann integrable (2) $f''(x)$ does not exist (3) $f'(x)$ is not continuous
0
votes
1answer
15 views

Show a polar function's diffrentiability

I need to show that $f(r,\theta)=r\sin(2\theta)\ r>0$ is differentiable at each point in its domain, and also decide whether it's $C^1$ or not. How should I approach this?
0
votes
1answer
31 views

Find equation of the tangent line at $\pi/3$

I need to find the equation of the tangent line to $f′(x) = 4 \sin x + 3 \cos x$ at $x= π/3$. I'm trying to incorporate the slope point formula. Progress This is what I got: $f'(x)= 4 \cos x- 3 ...
1
vote
0answers
12 views

What assumptions should be made?

take a problem like A trough is 12 feet long and 3 feet across. Its ends are isosceles triangles with altitudes of 3 feet. Water is being pumped into the trough at 2 cubic feet per minute. How fast ...
0
votes
2answers
47 views

Calculate the derivative of a power of $f$ in terms of $f$ and $f'$

(a) State precisely the definition of: a function $f$ is differentiable at a ∈ R. (b) Prove that, if $f$ is differentiable at a, then f is continuous at a. You may assume that $f '(a) = \lim {f(x) - ...
0
votes
2answers
13 views

Partial derivative of trig function

I need some assistance on the following calculus problem: Let $$w = 2\cot(x)+y^2z^2$$ $$x = uv$$ $$y = \sin(uv)$$ $$z = e^u$$ Find $\frac{\partial w}{\partial u}$ for $u = \frac{1}{4}$ and $v = ...
0
votes
1answer
29 views

The derivative of square root of $g$ from numerical values of $g$ and $g'$

How to do this: Function $g(x) > 0$, $g(1) = 9$, $g'(1) = 4$. If $h(x) = (g(x))^{1/2}$, find $h'(1)$ I got $2/3$. Is this correct?
0
votes
2answers
31 views

Why is this derivative not undefined at a given point?

I'm working on a problem from Keisler's Calculus (not homework, for my own amusement.) One of the problems is confusing me a bit. The first part goes like this: Suppose $g(x)$ is differentiable at $x ...
0
votes
0answers
30 views

Identifying f and a when given the formula for the derivative of f?

(Only need help with b) I tried to say that $f(a+h) -f(a) = (a+h)^{10}$ but I am getting nowhere. If $f(a+h)$ for $a=1$ is $(1+h)^{10}$, then $f(a)$ would have to be $0$ but then $f(a)$ would ...
1
vote
2answers
76 views

What do we mean by derivative of a function? What does it tell? [duplicate]

Taking the derivative of any kind of function is easy but I don't know why we take the derivative? Like $f(x)=x^2$ has the derivative $2x$, so what does it mean? I don't know how to define ...
0
votes
2answers
105 views

can you differentiate $y(x)=x^4 - 2x^2+8x$

Can you help me differentiate $$y=x^4 -2x^2+8x$$ with respect to $y$? Thank you.
0
votes
3answers
35 views

Differentiate Piecewise Functions

$$f(x) = \left\{\begin{array}{cl}x^3 \sin\frac{1}{x}, & x > 0\\ x \sin(x) & x \leq 0 \end{array}\right.$$ How do I find $f'(x)$? I tried using the definition of derivatives but it got me ...
1
vote
1answer
18 views

How to determine whether a piecewise function has a derivative?

Could someone show me a worked example of showing whether a piecewise function is differentiable at some $x=a$? I can show that it is continuous at $a$, as the limit as $x\to a$ (from both sides) ...
-1
votes
4answers
50 views

Derivative $ \frac{d}{dx} \ln(x+ \sqrt[]{ x^{2} + y^{2} }) $

$$ \frac{d}{dx} \ln(x+ \sqrt[]{ x^{2} + y^{2} }) $$ What I've done so far: $$1+\frac{0.5(x^{2})^{-0.5}2x}{x+\sqrt{x^{2}+y^{2}}}$$ $$1+\frac{\frac{x}{(x^{2})^{0.5}}}{x+\sqrt{x^{2}+y^{2}}}$$ ...
1
vote
0answers
28 views

Higher-order difference quotients

The Mean Value Theorem for Divided Differences says that if $f$ is $n$ times differentiable, and $x_0< x_1 < \dotsb < x_n$, then there is a point $\xi\in (x_0, x_n)$ such that $f[x_0, x_1, ...
3
votes
6answers
98 views

Find the first derivative $y=\sqrt\frac{1+\cosθ}{1-\cosθ}$

$$y=\sqrt\frac{1+\cosθ}{1-\cosθ}$$ my professor said that the answer is $$y'=\frac{1}{\cosθ-1}$$ she said use half angle formula but I just end up with ...
0
votes
5answers
91 views

Prove that $2^x+1$ is always greater or less than $3^\frac{x}{2}$?

There is any way to prove that for any real number $2^x+1 > 3^\frac{x}{2} $ or $ 2^x+1 < 3^\frac{x}{2}$ I tried using differentiation but it doesn't help any more due to $2^x$ and ...
1
vote
2answers
33 views

derivate of a piecewise function $f(x)$ at$ x=0$.

There is a piecewise function $f(x)$ $$f(x)= \begin{cases} 1 ,\ \ \text{if}\ \ x \geq \ 0 \\ 0,\ \ \text{if}\ \ x<0 \end{cases}$$ what is the derivative of the $f(x)$ at $x=0$? Is it $0$? Or ...
1
vote
2answers
38 views

derivative if a piecewise function

There is a piecewise function, $$ f(x)= \begin{cases} 0 & \text{if } x=0, \\ 1/x & \text{if } x \neq 0. \end{cases} $$ What is the derivative of this function at $x=0$ the txt says $+\infty$ ...
0
votes
1answer
17 views

Derivates and Limits in the Same Problem are an Issue.

I am working on the following problem:- Evaluate lim x→1 [( x^1/4 - 1 ) / ( x^1/3 - 1 )] by relating it to the derivatives of functions. Now this is quite a ...
0
votes
3answers
48 views

How to differentiate $-x^3(3x^4-2)$

What am I doing wrong? $-x^3*d/dx(3x^4-2)+(3x^4-2)*d/dx(-x^3)$ $-x^3(12x^3-2)+(3x^4-2)(-3x^2)$ $-12x^9+2x^3-9x^6+6x^2$ When just using the power rule it comes out to be $-21x^6+6x^2$
0
votes
2answers
41 views

Why cant we do substitution in differentiation but is ok in taylor series?

I have the same question 10 year ago when i was studying high school. I dont understand it and i give up the math. 10 year ago, i need to work with calculus during work and this question come to find ...
1
vote
3answers
81 views

Am I differentiating this wrong?

Differentiation is the opposite of Integration $$\begin{align}\int \cos^2x dx\end{align}$$ $$\begin{align}-\frac{\cos^3x}{3\sin x}\end{align}$$ Now if we differentiate $-\frac{\cos^3x}{3\sin x}$ we ...
0
votes
1answer
64 views

Prove $\sin(x)< x$ when $x>0$ using LMVT

According to Lagrange's Mean Value Theorem (LMVT), if a function $f(x)$ is continuous on $\left[a,b\right]$ and differentiable on $\left(a,b\right)$, then there exists some constant $c$ such that ...
0
votes
1answer
27 views

Derivative of $f(x)=\frac{7x^3+3x+30}{\sqrt{x}}$

$$f(x)=\frac{7x^3+3x+30}{\sqrt{x}}$$ $f^{\prime}(x)=\dfrac{\dfrac{1}{2\sqrt{x}}(7x^3+3x+30)-(21x^2+3)(\sqrt{x})}{(x^{1/2})^2}$ ...
1
vote
1answer
28 views

Derivative of $f(u)=\sqrt{8} \;u+\sqrt{6u}$

$$f(u)=\sqrt{8} \;u+\sqrt{6u}$$ $f(u)=\sqrt{8}\;u+(6u)^{1/2}$ $f^{\prime}(u)=\sqrt{8}+\dfrac{1}{2}(6u)^{-1/2}$ $=\sqrt{8}+3u^{-1/2}$ This was marked wrong, though. What am I doing wrong? ...
1
vote
1answer
20 views

If $A=5r^2$ and $\frac{dA}{du}=2$, what is $\frac{dr}{du}$

I am unsure exactly what this question is asking me to do. I think $\frac{dA}{dr} = 10r$ and I assume $u=a/2$ but I'm not sure where to go from there.
2
votes
1answer
30 views

Jacobian matrix of the inverse of a bijective function

Let $f:\mathbb{C}^n\rightarrow\mathbb{C}^n$ be a function such that $f=f(f_1,\ldots,f_n)$ and $f_i=f_i(x_1,\ldots,x_n)$. Also, $f$ is bijective and its Jacobian matrix exists. Does$f^{-1}\,$Jacobian ...
-5
votes
0answers
27 views

Use the alternative form of the derivative to find the derivative at x=c. [on hold]

Use the alternative form of the derivative to find the derivative at x=c. g(x) = (x=3)^(1/3), c=-3
0
votes
0answers
13 views

Calculus Single Variable: Find max and min of hard to graph function

Consider the function F defined by F(x)= integral from 0 to x of $t|sint(t)|dt$. Find the absolute maximum value and absolute minimum value of y=f(x). I know there's one at x= zero but the ones ...
0
votes
1answer
23 views

Evaluation of derivative: if $p(x)=b_0 + (x-z)q(x)$, then $p'(z)=q(z)$

I just wanted to confirm that I did this correctly, because this answer seemed too easy to obtain: $p(x)=b_0 + (x-z)(q(x)).$ Show that $p'(z)=q(z).$ My answer: $$\begin{align*} p'(x) ...
3
votes
3answers
49 views

If $f$ satisfies $\forall x\in\Bbb{R},0\leq f'(x), f''(x)$ and if $\exists a\in\Bbb{R}$ such that $0<f'(a)$, Then $lim_{x\to\infty}f(x)=\infty$

I got this problem: Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function that satisfy $\forall x\in\mathbb{R},0\leq f'(x)$ and $0\leq f''(x)$ Prove that if $\exists a\in\mathbb{R}$ ...
0
votes
1answer
41 views

Formal explication of $dx/dt = v(x)$ implies that $dt/dx=1/v(x)$

The tittle is all about my question. What is the formal explication of the fact that $dx/dt = v(x)$ implies that $dt/dx=1/v(x)$? Is that via geometry? analysis? differentiable forms? Can you give ...
0
votes
1answer
51 views

Second difference $ \to 0$ everywhere $ \implies f $ linear

Exercise 20-27 in Spivak's Calculus, 4th ed., asks us to show that if $f$ is a continuous function on $[a,b]$ that has $$ \lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2}=0\,\,\,\text{for all }x, $$ then ...
1
vote
2answers
35 views

Find the derivative of x^1/5 from the definition

I've been trying to figure out how to compute the derivative of $f(x) = x^{1/5}$ at $x=1$ from the definition. Here's what I've done: $$f'(1) = \displaystyle \lim_{\Delta x\rightarrow 0} ...
1
vote
1answer
47 views

There is a unique polynomial interpolating $f$ and its derivatives

I have questions on a similar topic here, here, and here, but this is a different question. It is well-known that a Hermite interpolation polynomial (where we sample the function and its derivatives ...
-3
votes
2answers
33 views

Implicit differentiation of $x^2+y^2=a^2$ [closed]

For the function $x^2+y^2=a^2$ show that $y'' = -a^2/y^3$ anyone know how to go about this one? Thanks in advance
-4
votes
0answers
31 views

Total Derivative - Application [closed]

I have the following question: The market price P of a used car in dollars is given by: P = 1000 + 0.005x - 4.00t Where X is the distance in miles west of a city and T is the time in days. If the ...
0
votes
4answers
53 views

If $f$ is continuous on $[0,\infty)$ and differentiable on $(0,\infty)$ and if $lim_{x\to\infty}f'(x)=0$ Then $f$ uniformly continuous on $[0,\infty)$

I got this problem: Let $f$ be a continuous function on $[0,\infty)$ and differentiable function on $(0,\infty)$ such that $\lim_{x\to\infty}f'(x)=0$. (1) Prove that for each $0<\epsilon$ there ...
0
votes
1answer
62 views

Differential and Integral calculus.

Can anyone here explain me, why do we take the Centre of mass of a conical shell using slant height and $dl$ whereas the centre of mass of a solid cone is calculated using the vertical height and ...
0
votes
1answer
59 views

What does $\ dx^2$ mean?

While writing the second derivative of y, $\frac{d^2y}{dx^2}$ what does the symbol $dx^2$ signify? I know that in case of the first derivative $dy$ means change in y and $dx$ means change in y and ...
0
votes
1answer
19 views

Find the absolute maximum and absolute minimum values of f on the given interval, f(x) = x^2 e^{-x/2}, [-2,8]

Here's the function: f(x) = x^2 e^{-x/2}, [-2,8] Sorry for asking this question again, but i cant seem to move forward. Can i get some help again? so i graphed the ...
0
votes
4answers
26 views

Finding the derivative of $v(r) = k(R^2 − r^2)$

The velocity (in centimeters per second) of blood r cm from the central axis of an artery is given by $$v(r) = k(R^2 − r^2)$$ where $k$ is a constant and $R$ is the radius of the artery. Suppose $k ...
1
vote
2answers
32 views

Algebraic issues with the calculation of the second derivative of $(a+be^x)/(ae^x+b)$

I'm trying to work out the 2nd derivative of $\dfrac{a+be^x}{ae^x+b}$ I have $f''=\dfrac{(ae^x+b)^2(b^2-a^2)e^x-2ae^x(ae^x+b)(b^2-a^2)e^x}{(ae^x+b)^4}$ There are so many terms, and I'm seriously ...
1
vote
2answers
29 views

How to find the derivative of improper integral with variable upper limit?

I have the integral from $-\infty$ to $y^2$ of the function $(e^{-|x|})$ and I need to find the derivative of this. That is, $$\frac{d}{dy} \int_{-\infty}^{y^2} e^{-|x|}\,dx$$ Usually derivative ...
-3
votes
2answers
40 views

How to find the derivative of $e(x) = \frac{x^2 + 80x + 40f}{rx}$? [closed]

Here $f$ and $r$ are constants. $$e(x) = \frac {x^2 + 80x + 40f}{rx}$$
1
vote
1answer
77 views

Does a word problem provide all information?

A while ago I asked a similar question about word problems and assumptions. Is it a definition or an accepted-fact that word problems provide all information about the relevant existence/situation in ...
2
votes
4answers
58 views

What is the rule behind this derivative?

$$\dfrac{\rm d}{{\rm d}t}\big(\sin^2(t)\big)=\sin(2t).$$ I don't understand what is the rule behind this derivation. I had tried to first rerivate sin() and then to derivate the square function, but ...
0
votes
1answer
28 views

Continuity and differentiability on piecewise function

Let $$f(x)=\begin{cases}x^2-3, & x<0;\\-3, & x\geq 0.\end{cases}$$ (a) Find the value of $x$ where $f$ is discontinuous (b) Find the value of $x$ where $f$ is non-differentiable ...
4
votes
5answers
275 views

Simple differentiation from first principles problem

I know this is really basic, but how do I differentiate this equation from first principles to find $\frac{dy}{dx}$: $$ y = \frac{1}{x} $$ I tried this: $$\begin{align} f'(x) = \frac{dy}{dx} & ...