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

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0
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3answers
54 views

Maximum value for $x(t)$ in $x(t) = -\frac{1}{2}gt^2 + vt$

In a book I am reading it says that the maximum value of $x(t)$ in $$x(t) = -\frac{1}{2}gt^2 + vt$$ is $\frac{v^2}{2g}$ and that this happens when $t=\frac{2v}{g}$ I cannot derive this though. When I ...
10
votes
1answer
275 views

Maxima of the function $\left \vert \int_{-1}^1 e^{i(ax+bx^2)}dx \right \vert$

I am looking for extrema of the function $$g(a,b):=\left \vert \int_{-1}^1 e^{i(ax+bx^2)}dx \right \vert$$ where $a,b >0$ are real parameters. I already plotted this function and got the ...
0
votes
1answer
47 views

Calculate the derivative of sin5x using limits

Well, that's it. How do you calculate $\frac{d}{dx} {\sin5x}$ using the limit formula for derivatives? $$\lim_{h \to 0} \frac {f(x+h)-f(x)}h$$ I managed to get a lot of sines and cosines using ...
1
vote
0answers
27 views

Extremal condition for series expansion coefficients

I want to maximize a coefficient in a series expansion, so the situation is the following. $f \in C^{\infty}$ and $f: \mathbb{R} \times \mathbb{R} \times [0,2 \pi] \rightarrow \mathbb{C}$. Now, we ...
1
vote
5answers
180 views

Is the function $\sin(\lvert x\rvert)$ differentiable?

It is readily shown that the function $\sin(\lvert x\rvert)$ is differentiable when $x\ne 0$. What I know is that $\lvert x\rvert$ is not differentable at $x=0$. But does $\sin(|x|)$ also follow the ...
18
votes
5answers
479 views

Geometric interpretation of mixed partial derivatives?

I'm looking for a geometric interpretation of this theorem: My book doesn't give any kind of explanation of it. Again, I'm not looking for a proof - I'm looking for a geometric interpretation. ...
1
vote
1answer
183 views

At what point does normal line intersect curve second time?

At what point does the normal line to $y=-5+4x+3x^2$ at $(1,2)$ intersect the parabola a second time? $y'=6x+4$ $m_{tangent}=6(1)+4=10$ $m_{normal}=-\dfrac{1}{10}$ $y=f'(1)(x-1)+f(1)$ ...
2
votes
1answer
144 views

Find $f$, such that $\,f,f',\dots,f^{(n-1)}\,$ linearly independent and $\,f^{(n)}=f$

I am trying to find a function $f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{C})$, satisfying the differential equation $$ f^{(n)}=f, $$ and with $\,f,f',\dots,f^{(n-1)}\,$ being linearly independent. ...
0
votes
3answers
30 views

Simple derivate question

In a paper I am reading about dynamics systems, they set the following variables: $a(\theta) = \ddot{\theta}$, $b(\theta) = \dot{\theta}^2$ Where $\dot{\theta}$ and $\ddot{\theta}$ are the first ...
0
votes
4answers
39 views

Derivative help [duplicate]

Use the definition of the derivative to find $f'(x)$ for $f(x)=\sqrt{x-2}$. The answer that I got was $$\frac{1}{2(x-2)^.5}$$. Is this correct? The second part asks use your answer from part 1 to find ...
0
votes
1answer
49 views

Use the definition of the derivative to find $f'(x)$ for $f(x)=\sqrt{x-2}$

Use the definition of the derivative to find $f'(x)$ for $f(x)=\sqrt{x-2}$. I don't even know where to start with this. I have done $\sqrt{x-2}= (x-2)^.5$. Is this correct?
0
votes
1answer
37 views

Is function differentiable?

Let $h: [0, \infty) \rightarrow \mathbb{R}.$ Suppose now that we have two integrable functions $f$ and $g$. If now the integrals $$\int_0^t h(t-x) f'(x) \ \mathrm{d}x \quad \mbox{and} \quad \int_0^t ...
0
votes
1answer
37 views

Derivative of SquareRoot with h-formula

I know the general formula for getting a derivative, and the formula for the derivative of the square root function, but I'm interested in how to do prove it using the formula for the definition of ...
5
votes
1answer
64 views

Non-differentiability of a function of two variables at a point

I have problems understanding this: Function $\;g(x,y)\;$ is given, for which a) $\;g_x(0,0)=7\;$ b) $\;g(t+2t^2,\sin3t+4t^2)=5e^t\;$ c) $\;\lim_{t\to 0}\frac{g(t,2t)-g(3t,4t)}t=10\;$ They ask ...
4
votes
1answer
91 views

A derivative which is not Lebesgue integrable on any interval?

If $f=x^2\sin(x^{-2})$, then $f'$ exists everywhere (including $x=0$) but $f'$ is not Lebesgue integrable on $[0,1]$ (precisely because of the singularity at $x=0).$ I'm trying to find a function $f$ ...
1
vote
2answers
106 views

Proving the differentiablity of a function.

Consider the differentiablity of the following function: $$f(x)=x\left(x+3\right)e^{-\frac{x}{2}}$$ My text proves the differentiability by taking 'Left Hand Derivative' and 'Right Hand Derivative' ...
0
votes
2answers
41 views

What are the tangents and asymptotes to $(x-1)(x+1)(x-3)$?

What are the tangents and asymptotes to $(x-1)(x+1)(x-3)$? The equation $$\frac{dy}{dx}=0$$ is not solvable so there are no tangents parallel to x-axis. The function is increasing and it has no ...
1
vote
0answers
121 views

Method for deriving an Exponential Moving Average?

I have a formula for an exponentially weighted moving average function defined recursively as: $S_t = a*Y_t+(1-a)*S_{t-1}$ Where: $a\in (0,1)\cap \mathbb{Q}$ $t$ represents time $Y_t$ is the value ...
0
votes
1answer
62 views

Constructing a sequence of function with bounded derivative

Let $f:\mathbb R\mapsto\mathbb R$ be a smooth function and analytic at $x=0$. I wish to find a sequence of functions $\{f_n\}$ such that $\{f_n(x)\}$ is convergent to $f(x)$ for all $x$ and $f'''_n$ ...
0
votes
2answers
82 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
21 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
51 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
20 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
0answers
61 views

Derivative and roots of polynomials

Given a polynomial $g(x)=\frac{f(x)}{(x-x_1)(x-x_2)}$, can it be proven that the roots of $g'(x)=0$ would lie in the interval $[x_1,x_2]$? Real/Complex, I'm not sure.
0
votes
4answers
81 views

Derivative of $f(x) = x^2 \sin(1/x)$ using the derivative definition

derivative of $f(x) = x^2 \sin(1/x)$ using the derivative definition When not using the derivative definition I get $\cos (1/x) + 2x \sin(1/x)$, which WolframAlpha agrees to. However when I try ...
0
votes
1answer
34 views

Differential equation with no nontrivial periodic solution

We are given $f=(f_1,f_2): \mathbb{R}^2 \rightarrow \mathbb{R}^2$, $C^1$ class with the property: $$(1) \ \ \ \forall_{(x,y)\in\mathbb{R}^2} \frac{\partial f_1}{\partial x}+\frac{\partial ...
3
votes
1answer
232 views

Finding continuity and differentiability of a multivariate function

Determine whether the following functions are differentiable, continuous, and whether its partial derivatives exists at point $(0,0)$: (a) $$f(x, y) = \sin x \sin(x + y) \sin(x − y)$$ ...
0
votes
1answer
27 views

Derivatives with functions of two or more variables

For the function $ln(4x^2+4y^2)$ when taking the derivative with respect to $x$, do you essentially leave the $y$ terms alone? I received the answer of $\frac{2x+y^2}{x^2+y^2}$, however the books ...
7
votes
1answer
99 views

Coincidence? : $d(ax^2+bx+c)/dx=\pm \sqrt{\Delta}$

As the title says, is it just a coincidence that $d(ax^2+bx+c)/dx=\pm \sqrt{\Delta}$? (where $\Delta=b^2-4ac$, i.e. discriminant of the quadratic). We can get this easily from rearranging the ...
0
votes
2answers
55 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) - ...
2
votes
2answers
96 views

differentiability of $\tan^{-1}(\frac{1}{|x|})$

How to justify, the following function is differentiable at origin or not? $f(x) = \tan^{-1}\frac{1}{|x|}$ if $x \ne 0$, $f(x) = \frac{\pi}{2}$ if $x = 0$. Even though mod x is not behaves well at ...
0
votes
2answers
33 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 = ...
1
vote
1answer
40 views

Differentiation of $xx^T$ where $x$ is a vector

How is differentiation of $xx^T$ with respect to $x$ as $2x^T$, where $x$ is a vector? $x^T $means transpose of $x$ vector.
0
votes
1answer
35 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
38 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
34 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 ...
0
votes
0answers
31 views

derivative $\frac{d}{dt} \left[ \frac{-2}{\dot{x}^3} + \frac{-2x^2}{\dot{x}^3} \right]$

I need to find the derivative of the following equation: $\frac{d}{dt} \left[ \frac{-2}{\dot{x}^3} + \frac{-2x^2}{\dot{x}^3} \right]$, where x and x' are fuctions of time. As an example, an easier ...
0
votes
0answers
24 views

Expand function using Maclaurin's series(infinite form)

Expand the function f(x)=log(1+x) in powers of x in an infinite series stating the validity of such expansion for x belonging to (-1,1]. The question actually asks to show that cauchy's remainder or ...
1
vote
2answers
92 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
114 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
45 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
52 views

How do I evaluate this implicit differentiation?

I was reading a Classical Mechanics book. The author was deriving Kepler's equation. He was changing variables for integrating the stuff later. Here's my reproduction of that figure. $N$ is the ...
1
vote
1answer
40 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
55 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
32 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, ...
4
votes
6answers
163 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 ...
1
vote
3answers
73 views

Unable to differentiate $\cos(x) \cos(2x) \cos(3x)$ and $\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$

I apologize for the lack of LaTeX. I will update this question with the proper LaTeX as soon as possible. I am having trouble with two differentiation exercise questions and was hoping someone could ...
0
votes
2answers
37 views

Euler-Lagrange equation: Differentiation with respect to x

I got stuck in my lecture notes after a supposed differentiation of the Euler-Lagrange equation: $$\dfrac{\partial f}{\partial y}-\dfrac{d}{dx} \left( \dfrac{\partial f}{\partial y'}\right) = 0$$ ...
0
votes
5answers
100 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
0answers
75 views

Calculate D(f o g)(1,2)

I'm doing this problem: Let: $g:\mathbb{R}^{2} \rightarrow \mathbb{R}^2$ and $f:\mathbb{R}^{2} \rightarrow \mathbb{R}^2$ be a differentiable function such that: $g(0,0)=(1, 2); \ \ g(1,2)=(3,5); \ \ ...