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

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Maximize profit

my book (George F. Simmons - Calculus with analitic geometri) have the following question: An library could buy from the book publisher the book "Rituals" with a cost of 40.0 each. The manager from ...
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1answer
25 views

Is it possible to have a inflection on a vertical asymptote?

I found the derivative of a function to be f'(x)=8/x^3 and thus its second derivative as f''(x)=0/3x^2. After setting the second derivative to zero and doing the substitution into the parent function, ...
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2answers
42 views

Can an inflection exist if there's no max/min?

Very quick question: if a function doesn't have a maximum nor minimum, can it still have a point of inflection? I believe that these two go hand in hand and without one you can't have the other but ...
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47 views

How many continuous functions are differentiable? [duplicate]

Consider the set of continuous functions $\mathbb{R} \to \mathbb{R}$. I assume that the subset that are not everywhere differentiable accounts for almost all of them. Is this true? What is the precise ...
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3answers
56 views

Derivative of $\sqrt{x^2+1}$

Ive been given this rule and asked to differentiate $\sqrt{x^2+1}$, however I am not sure what I am missing.It is said that if f is differentiable at x and f(x)>0. ...
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2answers
17 views

Matricial differentiation $x x^{\top} b $

What is the drivative of $x x^{\top} b $ with respect to x, knowing that b is constant vector?
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6answers
44 views

Using Chain Rule and Product Rule to find derivative

I have to find the derivative of the following function: $$f(x) = (x^3+ 4)(4x^5 + 2x − 5)^{1/2}$$ To start solving this, I've dissected the equation and realize that I must use the product and chain ...
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1answer
31 views

Help in differentiation [on hold]

Can somebody please show the steps of how differentiation of Shannon's entropy yields the following result? $H = -\sum_{l=0}^{L-1} p(l)\log_2[p(l)]$ The result of differentiating is $H_m = ...
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24 views

Can all null-homotopy be made differentiable on arbitrary metric space?

Let $M$ be a metric, and assume that it is simply connected. For a closed curve $f$, we define it to be differentiable iff for any $x$ then $\lim\limits_{h\rightarrow 0}\frac{d(f(x),f(x+h))}{h}$ ...
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1answer
103 views

Derivatives in the real world

Two row boats start at the same location, and start traveling apart along straight lines which meet at an angle of $\pi/3$. Boat A is traveling at a rate of $10$ miles per hour directly east, and boat ...
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0answers
38 views

Taking derivative under the integral sign

Reading a textbook and stuck on this one detail... would like to confirm my understanding. The book defines a function $\eta \in C^1(\mathbb{R})$ satisfying $0 \leq \eta \leq 1$, $0 \leq \eta^\prime ...
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2answers
33 views

derivatives of a vector of functions with respect to a vector

Let $\vec W \in \mathbb R^3$. What is the general solution to: $$\frac{\partial}{\partial \vec{W}} \begin{pmatrix} f(\vec W) \\ g(\vec W) \end{pmatrix} $$ I think that in the ...
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1answer
23 views

How is the power rule applied to whole numbers

For the following function, how does the $+1$ become $0$ when finding its derivative via the power rule? Original function: $f(x) = 6x^2 − 4x^{-1} + 5x^{-2} − 2x + 1$ Derivative: $f '(x) = 12x + ...
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1answer
42 views

Chain rule application in fundamental Theorem of Calculus

I have attached a question that I came across in trying to understand the fundamental theorem of calculus. The solution (highlighted with an arrow). I have difficulty understanding why the chain rule ...
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1answer
31 views

Total derivative proof [on hold]

The wikipedia article does not prove it http://en.wikipedia.org/wiki/Total_derivative Neither the top articles in google search. Could somebody help me proving it? I've found this: ...
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1answer
517 views

Where is the error in my proof that all derivatives are continuous?

I know that this can not be true due to counter-examples but I don't know where the error in my reasoning is. Assumption: If $f(x)$ is differentiable in $\mathbb{R}$ then the derivative $f'(x)$ is ...
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3answers
195 views

Determining the maximum value for the solution of this delay differential equation?

I am working on the following delay differential equation $$\frac{df}{dt}=f-f^3-\alpha f(t-\delta)\tag{1},$$ where $\frac{1}{2}\leq\alpha\leq 1$ and $\delta\geq 1$. I know that there are three ...
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1answer
60 views

Simplify $\dfrac{3}{49 \sqrt[7]{x^4}} -5x^4$ [on hold]

From a derivative my professor said it's not yet simplified. how? i think this is it already.
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29 views

A question on limits

$$\lim_{h\rightarrow 0}\frac{2^{8\cos(h)}}{8h}\left [ sin^{8}(\pi/6+h))-sin^{8}(\pi/6) \right ]$$ MY ATTEMPT: for $\lim_{h\rightarrow ...
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1answer
12 views

Equation of a line with a positive gradient [on hold]

Two straight lines passing through the point (0,2) are tangent to the graph of the function y=1-x^2. Find the equation of the line with a positive gradient.
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3answers
208 views

Find all values of $x$ at which the tangent line to the given curve has intercept $ y= 2$

Find all values of $x$ at which the tangent line to the given curve has intercept $y = 2$ I am confused about the $y$-intercept $2$ the function $$f(x) = \frac{(2x + 5)}{(x + 2)}$$ The derivative ...
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0answers
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how do calculus this derivate? [on hold]

how do calculus this derivative? $$f(x) = (2x-x^2)^{1/2}$$ And how do I calculus this derivative? $$F(x) = (\sin{x}/(1+\cos{x}))^2 $$
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2answers
45 views

What and how do derivate? [on hold]

How do I derive this function? $f(x) = x(e^{-x^2})$ I need the first and second derivative.
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2answers
40 views

How do you answer these questions regarding the Taylor series method?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the Taylor series method. (b) Assuming $f(x)\in C^3$, evaluate the ...
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1answer
43 views

Measuring sums of complex alternating series

Suppose we have functions $$f(x) = \sqrt{x}, \space g(f) = \frac{df}{dx}+\frac{d^2f}{dx^2}+\frac{d^3f}{dx^3}\space ...$$ Applying function f(x) to g(f) we get: $$g(f(x))=\frac{1}{2}x^{-\frac{1}{2}} - ...
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23 views

How do you solve part (b) to this polynomial interpolation question?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the polynomial interpolation method. (b) Assuming $f(x)\in C^3$, evaluate ...
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3answers
47 views

How to evaluate $\lim_{x \to \infty}\left(1 + \frac{2}{x}\right)^{3x}$ using L'Hôpital's rule?

I'm stuck on how to evaluate the following using L'Hôpital's rule: $$\lim_{x \to \infty}\left(1 + \frac{2}{x}\right)^{3x}$$ This is a problem that I encountered on Khan Academy and I attempted to ...
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1answer
22 views

What is the Jacobian of the following function

Consider a function F: $R^n \to R^n$ defined by $$f(u) = A*u*(n+1)+\lambda *B$$ Where A is a tridiagonal n-by-n matrix with -2 on the main diagonal and 1 on the off diagonals. B = $\begin{pmatrix} { ...
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3answers
31 views

Factoring when differentiating expressions

I'm having trouble with differentiating a expression. I do it one way, wolfram alpha does it another. Let me show you what I mean. The original expression is this: $$\frac{1}{2u^3}$$ I start by ...
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30 views

Differentiability of polynomials

Trivial question but I am confused with the notation If $p_{n-1}$ is a polynomial of degree $n-1$, is it $\in$ the differentiability class C^n$? Obviously if $p_n$ is a polynomial of degree $n$, ...
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4answers
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Implicit Derivative approaches

Sorry for my excessive verboseness... Here's the equation as given: $$x = 10 + \sqrt{x^2 + y^2}$$ Here are my direct implicit steps without modifying original equation: $$\eqalign{ \dfrac{\mathrm ...
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1answer
26 views

Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds. (Cf. discussion on p. 45.) Also note that though I were able to resolve the first problem the second one is still ...
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3answers
65 views

How to find the values of m and b?

How do I find the values of m and b when: a) the function is continuous in $x = \pi$ b) the function can be derivated in $x =\pi$ $$y=\begin{cases} \sin x & x<\pi \\ mx+b & x\ge ...
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1answer
53 views

Finding inflection points using the second derivative

So I have the function $y=(1+x^2) e^{-x^2}$ I find its first derivative $y'=-2x^3e^{-x^2}$ and its second derivative is $y''=e^{-x^2}(-6x^2+4x^4)$. Then I find the roots of $y''$ and they are $0, \pm ...
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2answers
206 views

Differentiation, from first principles

I am having problems with this question, it would be wonderful if someone can help. Given that $f(x)= x^2 + x - 3$ 1) Find $f(x + h)$ 2) Then express $f(x+h)-f(x)$ in its simplest form 3) Deduce ...
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2answers
20 views

Global maximum and global minimum a combination of values

I have two variables $x$ and $y$. I can have them both in any combination of positive numbers that will add up to $1000$ and need to find the combination in which $z$ is at its minimum in the ...
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1answer
30 views

Polynomial and its derivative have a common factor?

When is $gcd(p(x),p'(x))\ne 1$ where $p(x)$ is a polynomial? That is when does the derivative of a polynomial and the polynomial has a common factor? By when i mean some condition for the ...
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1answer
30 views

Derivative of logistic loss function

I am using logistic in classification task. The task equivalents with find $\omega, b$ to minimize loss function: That means we will take derivative of L with respect to $\omega$ and $b$ (assume y ...
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24 views

Please help me check this derivative work

I have $$ J_{\theta}(X) = - \frac 1 m \cdot \left[ y \cdot ln( h_{\theta} (X ) ) + ( 1 - y) \cdot ln ( 1 - h_{\theta}(X) ) \right] $$ I need $\frac d {d\theta} J_{\theta}(X)$. I tried many time, and ...
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68 views

Related rates of change: distance between a plane and a car

A highway patrol plane flies $3 m$ above a level, straight road at a constant speed of $120 m/s$. The pilot sees an oncoming car and with radar determines that at the instant the line of sight ...
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2answers
101 views

Derivative of $(-2)^{x+1}$ [on hold]

Can we compute the derivative of $(-2)^{x+1}$? This may sound silly, but think about it. We cannot apply any of our formulae on it. I think we may have to go old school with this one
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1answer
23 views

Sign convention for derivatives in a $\mathbb{Z}_2$ graded space

Suppose $V=V_0\oplus\theta V_1$ is a $\mathbb{Z}_2$ graded super vector space. Note: Since $\theta^2=0$, this implies $\theta\mathrm{d}\theta=-\mathrm{d}\theta\cdot\theta$. However, I wish to know ...
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2answers
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How to find derivative of an integral of this type

$$f(x) = \int _x^{e^x}\:\left(\sin t^2\right)\,dt$$ How to find the derivative $f'(x)$ Attempt: $\sin (e^{x^2}) e^x$
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1answer
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Math question: Calculus [on hold]

"A rancher would like to enclose two adjacent rectangular corrals that cover a total area of $12,000 ft^2$. If material for the fence costs $3.5$ usd per foot, find the dimentions (length and width) ...
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How to solve this problem about production and derivatives? [on hold]

If p(x) is equal to the production of a factor when there are x workers, then the average productivity of the work force is: A(x) = p(x)/x a) Find A´(x). Why does the factor need to hire more ...
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Derivative of a generalized hypergeometric function

Let $$f(a)={_2F_3}\left(\begin{array}c1,\ 1\\\tfrac32,\ 1-a,\ 2+a\end{array}\middle|-\pi^2\right).$$ How to find $f'(0)$ in a closed form?
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2answers
69 views

Gradients and functions on matrices

Given a twice differentiable $f: \Bbb R \to \Bbb R$, with continuous second order derivative. We define $$F(x) = \sum_{i=1}^{m}f(x_i)$$ and $$L(x) = \sum_{i=1}^{m}f( \langle a_i, x \rangle+ b_i),$$ ...
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69 views

Matrix exponential Differentiation

We have the equation $e^X = \sum_{k=0}^\infty{1 \over k!}X^k.$, where X is a matrix of dimension $3 \times 3$ . Now I have a function $f(x)=C_1x+C_2*\frac{x^2}{2} $ where $C_1,C_2,f(x)$ has ...
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27 views

First derivative of this secial function

What is the derivative of the following function: $$f(x) = \frac{a}{((\sqrt{b+bx})(d-\sqrt{e+gx}))^2}$$
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1answer
20 views

Using convolution to impose differentiablilty.

If I had a function $g$ that was not differentiable at a known point, is it possible to convolute it with say a $C^{\infty}$ function $f$, resulting in a differentiable function? Thanks in advance!