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

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1answer
32 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
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1answer
20 views

does the following function have all directional derivatives?

$$xy\sin(\frac{1}{xy})$$ the function has partial derivatives at every point , but i wanted to know whether this function had directional derivatives at every point? for $x=0$ the function is ...
1
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2answers
67 views

Finding example of a special type of continuous differentiable function

Give example of a continuous function (if exists) $f : [a,b]\to \mathbb R$ differentiable in $(a,b)$ such that $f(a)f(b) \ne 0$ , the set $A:=${ $x \in (a,b) : f(x)=0$ } is infinite but not an ...
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1answer
30 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 $$ \lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(0)}{h^2}=0\,\,\,\text{for all }x, $$ then ...
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2answers
31 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} ...
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0answers
21 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 ...
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0answers
7 views

How do you find the gradient between two different curves that passes through an arbitrary point between the curves?

Given a graph like this, XC, and ZC is it possible to find YC, and if so, how? fB(x) and fA(x) are some known but different functions at ZB and ZA, respectively. fC(x) is NOT a known function but ...
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0answers
19 views

proof of derivative of a complex function

suppose $u(x,y)$ is harmonic in a domain $D$ and $v(x,y)$ is an harmonic conjugate of $u$. Let $f(z)=u(x,y)+iv(x,y)$. Prove $f'(z)=u_x+iv_x$.
-3
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2answers
31 views

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

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
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0answers
16 views

Frechet Derivatives, how to obtain this inequality (i think thats easy)

If $(e_1,e_2,\ldots,e_n)$ is the canonical base of $R^n$ and if $\alpha=(\alpha_1,\ldots,\alpha_n)$ is a multi-index of $N^n$, we have that $\partial^\alpha v(x)= ...
-4
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0answers
29 views

Total Derivative - Application [on hold]

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 ...
1
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0answers
23 views

An example of the solutions for Second order differential equation

This question is the simpler version from here. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ ...
-2
votes
1answer
35 views

Showing that a derivative is positive for all x? [on hold]

I have a function with the domain [a,b]. I also have its derivative. How do I show that this derivative is positive for all x in [a,b]? Basic question, I know.
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2answers
22 views

How to solve when the unknown is given?

A curve has a gradient function $px^2 - 5x$, where $p$ is a constant . The tangent to the curve at the point $x=1$ is parallel to the straight line $y+2x-5=0$. Find the value of $p$.
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4answers
45 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
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1answer
49 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 ...
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0answers
26 views

How proceed from here with leibnitz theorem for nth derviative, for logx/x

I am trying to solve these questions , I am in freshman year.Can someone please tell me how to get started with question no:12. I have no idea where to begin with. Rewriting as mentioned in ...
0
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1answer
56 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 ...
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3answers
43 views

Finding the values of $a$ and $b$ such that $f$ is continuous and differentiable at $x = 1$? [on hold]

The equation is $F(x) = \begin{cases} x^2 & \text{if } x \leq 1 \\ ax+b & \text{if } x>1 \end{cases}$ Differentiable at $x = 1$ I'm having a hard time understanding on how to ...
2
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3answers
56 views

$\displaystyle \frac{d}{dx}2^x$ where $x=0$

I put into Wolfram Alpha: d/dx 2^x Where it told me $f'(x)=2^x\log(2)$. Then I put in d/dx 2^x where x=0 and it said "$\displaystyle \log(2)\approx0.693147$" I know through Wolfram ...
0
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1answer
35 views

Integrals involving roots

I am bit stucked with an integration form while doing one of my proofs for a graphics application.Issue is I cant take out the terms from the trigonometric functions for a proper known integral ...
0
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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 ...
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4answers
24 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
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2answers
28 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 ...
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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 ...
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2answers
35 views

Can the derivative of an absolutely continuous real function have a simple discontinuity?

If $f'$ exists everywhere, then we know that it cannot have any simple discontinuities. But in this case we only know that $f'$ exists a.e. (since $f$ is absolutely continuous). More specifically, ...
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2answers
36 views

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

Here $f$ and $r$ are constants. $$e(x) = \frac {x^2 + 80x + 40f}{rx}$$
2
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1answer
58 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
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4answers
54 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 ...
2
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0answers
44 views

If the set of values , for which a function has positive derivative , is dense then is the function increasing ?

Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $A:=${ $x \in \mathbb R :f'(x)>0$ } is dense in $\mathbb R$ , then is it true that $f$ is an increasing function ?
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1answer
11 views

On the existence of a non-constant sequence whose differentiable image converges [duplicate]

Let $f: [a,b] \to \mathbb R$ be a function differentiable in $(a,b)$ , then is it true that there is a non-constant sequence $(x_n)$ in $(a,b)$ such that the sequence $\big(f(x_n)\big)$ is ...
0
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2answers
24 views

Where am I wrong with this derivative?

I want to derivate this function : $$f(t) = \frac{3}{\sin(t)}$$ I know that the derivative of $\frac{u(x)}{v(x)}$is$\frac{u'v-uv'}{v^{2}}$ in general and that in this fraction : $$u'(t) = 0$$ $$v'(t) ...
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0answers
49 views

Impatience and interest rate

I'm having difficulties solving the following problem in economics. I come from a mathematical background, and it's hard for me to get some of the terms: Consider a two-period economy with a ...
0
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1answer
48 views

Interpretation of differential form

We know what is the interpretation of a total differential, ex.: $$df=\frac{\partial f}{\partial x} dx+ \frac{\partial f}{\partial y} dy$$ But what is the interpretation of a 1-form and its exterior ...
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0answers
10 views

Partial derivative within domain

Having trouble understanding how this partial derivative is calculated within a domain: $$ max (a-a^2-0.5ac-0.5ad) \text{ for a, where 0<a<1} $$ Doing it myself (and ignoring the domain) I ...
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0answers
13 views

Differential and a notation problem

Let $df(x) = f'(x)\,dx \:\:\:\:\: (1)$ Now we want to integrate both sides and we get: $f(x) = f(x)$ But now we want to differentiate again and we get $f'(x)=f'(x)$ I just don get it. If we say ...
1
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1answer
38 views

Differentiating a Quadratic Form

I'm having some trouble differentiating a quadratic form. I'm tasked with showing that $P(x) = \frac{1}{2} \left(b-Ax\right)^T C (b-Ax)$ is minimized by a vector $x$ satisfying $A^T C A x = A^T C b$. ...
0
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1answer
19 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 ...
3
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1answer
58 views

Finding $\int_{0}^{1} \frac{\log(1+x)}{1+x^2} {\rm d}x$ by differentiating under the integral sign.

I've tried to find this integral by the method already outlined in the title. I decided to let $$ \displaystyle I(\alpha) = \int_{0}^{1} \dfrac{\log(1+\alpha x)}{1+x^2} \text{ d}x. $$ From this ...
0
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1answer
12 views

What is the derivative of a Radial Basis Interpolation function?

A radial basis interpolation function is described as: $ f(\textbf{x})=\sum_{k=1}^N c_k \phi(\lVert \textbf{x}-\textbf{x}_k \rVert_2), \ \textbf{x}\in\mathbb{R}^s $ where $\textbf{x}_k$ are the $N$ ...
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5answers
268 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} & ...
3
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0answers
114 views
+300

Integration of product of functions

Sir, I have been doing a proof related to one research topic. But after a long effort, I got ended up in a messy integration equation. Could you give me some suggestions to solve this equations? (Any ...
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0answers
24 views

Integration with matrices

I have written two equations in matrix format as follows $m(t)={\begin{pmatrix} 200\\ 300\\ 400\\ 500 \end{pmatrix}}^T \begin{pmatrix} ...
2
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3answers
57 views

If $f$ is continuous on $[a,b)$ and differentiable on $(a,b)$ such that $\lim_{x\to b^{-}}f(x)=\infty$, Then $f'$ is not bounded above in $(a,b)$.

I got this problem: Let $f$ be a continuous function on the interval $[a,b)$ and differentiable on the interval $(a,b)$, Prove that if $\lim_{x\to b^{-}}f(x)=\infty$, Then $f'$ is not bounded above ...
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0answers
18 views

Prove - Integrate/Differentiation of Normal Distribution [on hold]

Can someone prove this? $\dfrac{\partial}{\partial x}\left(\displaystyle\int_{-\infty}^{2x}\dfrac{1}{\sqrt{2\pi}}e^{-\frac{(2x)^2}{2}}dx \right)=\dfrac{1}{\sqrt{2\pi}}e^{-\frac{(2x)^2}{2}}\cdot ...
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0answers
9 views

Negative Gaussian Log Likelihood Minimization - question about scaling and Hessians

I received this question from my niece, and had to admit it's quite over my head. I wonder if anyone here could assist? I am doing a minimisation with the negative of a gaussian log likelihood ...
0
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3answers
54 views

Range of function with limit?

I have a function $$f(x)=\frac{2x^2 - x - 1}{x^2 + 3x + 2}$$ from the interval $[0,\infty)$ The limit of this function is $2$. Is the range then simply from $f(0)$ to $2$, and if yes, would I ...
1
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0answers
35 views

Differentiation of the Law of Cosines, where a, b, c, A, B, and C are functions of time t

Is the differentiation of the law of cosines ($c^2= a^2 + b^2 - 2ab\cos C$) this? a, b, c, A, B, and C are functions of time t. $$2c \frac{dc}{dt} = 2a \frac{da}{dt} + 2b\frac{db}{dt} - 2b \cos C ...
0
votes
2answers
51 views

Simplification & Differentiation of $\frac{2x}{x^{1/3}}$

Above is the image I had taken a snap shot of. I was working on the problem # 24. I got to rewrite the function as: $y = 2x(x^{-1/3})$ I differentiated it and got the $y'$ as: $y' = ...
1
vote
2answers
35 views

Implicit differentiation with logarithm

Find $y'$ if $y=\ln(7x^2+3y^2)$. I'm kind of confused on this problem and could use everyone's help. Am I supposed to take the derivative first, which is $y=\ln(14x+6y)$? If so, how do I go from ...