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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2answers
33 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
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1answer
15 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 ...
4
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0answers
35 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 ...
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3answers
38 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 ...
-1
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0answers
24 views

derivative of a symmetric matrix

Let $\mathbf{A}\in\mathrm{C}^{m\times n}$ be a random matrix. Now we define, $\mathbf{X} = \mathbf{A}\mathbf{A^*}$. Where, $\mathbf{A^*}$ is complex conjugate of $\mathbf{A}$. Thus, $\mathbf{X}$ is a ...
1
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0answers
26 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, ...
0
votes
2answers
11 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
33 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.
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1answer
25 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?
1
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1answer
35 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 ...
0
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2answers
29 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
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0answers
29 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
1answer
335 views

Maximizing cross sectional area of trapezoid

The task is to fold a piece of sheet metal that measures 60 cm across in such a way as to form a trapezoidal "gutter" (a trough for carrying rainwater) with the maximum possible cross-sectional area. ...
0
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0answers
22 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 ...
1
vote
1answer
31 views

Differentiation under the integral sign (one complex variable)

Let $u(z), u'(z)$ be complex-analytic functions on an open neighborhood $\Omega \subseteq \mathbb{C}$ of the origin. Also, let $f(X)$ be a complex-analytic function. For $s \in [0,1],$ define $$g(s,z) ...
0
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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 $-21x^6+6x^2$
1
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2answers
74 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 ...
-1
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0answers
37 views

derivatives and integers

I have a very fundamental question, We are given that $f(x_1)\leq a, f(x_2) \geq b$ (for distinct integers $x_1,x_2$), then there exists $y \in [x_1,x_2]$ such that $f'(y) \geq b-a$. (Firstly is the ...
0
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0answers
12 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 ...
0
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3answers
34 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 ...
3
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6answers
95 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
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2answers
96 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.
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4answers
48 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
1answer
16 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) ...
0
votes
5answers
90 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 ...
0
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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 ...
1
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4answers
48 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 ...
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0answers
58 views
+100

Existence of a bounded function satisfying a second order differential equation

This question is a variation version from here. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ ...
0
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2answers
21 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: $$\dfrac{\partial f}{\partial y}-\dfrac{d}{dx} \left( \dfrac{\partial f}{\partial y'}\right) = 0$$ ...
2
votes
1answer
628 views

Minimizing L1 Regularization

I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
1
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0answers
19 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); \ \ ...
1
vote
2answers
31 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 ...
0
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0answers
12 views

forward backward rules of Euler for slope

I am asked compute the slope of a trajectory at two points using the "forward/backward rules of Euler." This is a time-sampled function with regular intervals. I'm having a little bit of trouble ...
1
vote
2answers
35 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
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2answers
16 views

Derivatives and Differentiables [on hold]

Where is the greatest integer function $f(x) = \lfloor x \rfloor$ not differentiable? Find a formula for $f'$ where it is defined?
0
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2answers
27 views

Is there an easy way to study the sign of this?

I would like to study the sign of this derivate, but I don't know where to start : http://www.wolframalpha.com/input/?i=derivate+sqrt%28x%5E4-7x%5E2%2B16%29
1
vote
1answer
39 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 ...
1
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1answer
84 views

A polynomial agreeing with a function and its derivatives

If we want $$p(x_i)=a_i, \qquad x_1 < \dotsb < x_{n+1},$$ then there is a unique polynomial of degree $\leq n$ that accomplishes this (Lagrange interpolation). If we want $$p(x_i)=a_i, \qquad ...
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2answers
80 views

Modified Hermite interpolation

I asked similar questions here and here, but I tried to formulate this one in a sharper way. Is anyone aware of some literature on polynomial interpolation where we sample the function and its ...
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2answers
39 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
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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
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1answer
57 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
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1answer
26 views

demonstrate that function is increasing in intervals that are multiples of pi?

I have the derivative: $$- \frac1{x^2} + 1 + \frac{\cos^2(x)}{\sin^2(x)}$$ and am supposed to show that this is positive for all $x \in (n\pi, (n+1)\pi)$. How exactly am I supposed to do that? ...
3
votes
1answer
134 views

Problem with notation in a thesis

I am struggling with section 3.3 of the following thesis https://smartech.gatech.edu/xmlui/bitstream/handle/1853/29610/grigo_alexander_200908_phd.pdf. Page 21 is fine, then the problems occur in ...
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
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0answers
20 views

Change order between integral and differential calculation

Are those right? And I want to ask, in general case, when we can change the order of diff and integral: diff(integrate(L(x,y))) integrate(diff(L(x,y)))
1
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1answer
39 views

What to do *rigorously* when the second derivative test is inconclusive?

How do you rigorously check if a point is a local minimum when the second derivative test is inconclusive? Does there exist a way to do this in general for arbitrary smooth (or analytic...) functions? ...
0
votes
1answer
25 views

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

$$f(x)=\frac{7x^3+3x+30}{\sqrt{x}}$$ $f^{\prime}(x)=\dfrac{\dfrac{1}{2\sqrt{x}}(7x^3+3x+30)-(21x^2+3)(\sqrt{x})}{(x^{1/2})^2}$ ...
2
votes
0answers
10 views

Proof that maximal interval of existence exist and bounded

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
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? ...