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

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1answer
33 views

Schwarz' theorem for real and imaginary part of a function

Let $U\subset \mathbb{C}$ open, $f:U\to\mathbb{C}$ has continuous second partial derivatives in $U$, then Schwarz' theorem states that in U it is $$\frac{\partial^2 f}{\partial x\, \partial ...
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1answer
45 views

Derivative of the integral with respect to the function

Consider this function: $$ E[L] = \int\int\{ y(x) - t \}^2p(x,t)dx dt $$ I try to figure out how to take the derivate of this function with respect to $y(x)$. In the book it is: $$ \frac{ \delta E[L] ...
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2answers
39 views

Taylor Series and Differentiation with Sigma notation $f(x) = \frac{x}{(2-3x)^2}$

Use Term By Term Differentiation to Find the Taylor Series about $x$=3 for Give The Open Interval of Convergence and express as sigma notation $\sum A_n(x-3)^n$ $f(x) = \frac{x}{(2-3x)^2}$ So I ...
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0answers
82 views

partial derivative of a facet normal wrt to one of its vertex

I am struggling to understand the derivation of an equation in a paper (A Bayesian Method for Probable Surface Reconstruction and Decimation, specifically Eqn. 16). Basically they define three ...
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1answer
70 views

Second Derivative of Cusp

I know the first derivative does not exist at a cusp. Does this statement also hold for the second derivative?
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2answers
42 views

Increasing Function $\implies$ Positive Derivative

If $f$ is increasing in $[a,b]$ and is differentiable in $(a, b)$, then $f'(x)>0$ in $(a, b)$. My thoughts: Even if a function is increasing, it can be increasing on the $2$nd and $3$rd quadrant, ...
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0answers
9 views

derivative of Neumann boundary condition

Consider $\Omega\subset\mathbb{R}^N$ and $\Gamma$ its boundary. If a function $v$ in $\Omega$ such that $v=0$ and $\frac{\partial v}{\partial n}=h$ on $\Gamma$ with $n$ being normal vector, then how ...
2
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1answer
283 views

Cusps and Points of Inflection

Can cusps be considered points of inflection? I'm getting conflicting information but my thought process is that cusps cannot be points of inflection? Can points of inflection exist when there is a ...
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3answers
58 views

Derivative function of $x/\ln(x)$

I was trying to solve the following limit, $$\lim_{x \to 1^+} \frac{x}{\ln(x)}$$ using L'Hopitals rule, but when applying the limits to the given equation, the function turns to the form ...
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2answers
18 views

Derivatives of quotients (rate of change)

PRODUCT RULE ONLY A function modelled $D(t)=.5(t^2+8)(t+4)$ where t is years from now. Find the rate of change when $D(t)=756$ I did most of the work already bringing 756 to the other side and ...
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0answers
25 views

Proving the differentiability and continuity of $ f(t)=\frac{1}{it(b-a)}(e^{itb}-e^{ita})$ extended by $f(0)=1$

Proving the differentiability of: $$ f(t)=\begin{cases} \frac{1}{it(b-a)}(e^{itb}-e^{ita}),&t\neq0 \\ 1, &t= 0 \end{cases}$$ This is the characteristic function of $\mathcal U(a,b)$ and the ...
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2answers
30 views

If a function has $k+1$ roots on an interval, then its $k$th derivative has a root there

Suppose that $$a \le x_0 < x_1 < ... < x_k \le b,$$ $$f(x_0) = f(x_1) = ...= f(x_k) = 0,$$ and $$f(x), f'(x),...,f^k(x),$$ are all continuous on $[a,b]$. Show that there is a $\delta \in ...
2
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0answers
17 views

Partial Differentiation on the Wave Equation [duplicate]

Consider the equation $$\frac{d^2u}{dt^2}=c^2\frac{d^2u}{dx^2}$$ where c is some constant. Define new variables, $\sigma=x-ct $ and $\gamma=x+ct$. Now show that the equation becomes ...
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2answers
49 views

Why $\frac{d^2u}{dx}=2u\cdot \frac{du}{dx}$

I don't see why the below expression is correct: $\frac{d^2u}{dx}=2u\cdot \frac{du}{dx}$ I believe it's rather easy, but I can't see how it is correct. Can anyone tell me why the above equation is ...
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2answers
24 views

Derivative of $\cos^{-1} (1-\tfrac{h}{r})$ with respect to time

I'm trying to solve a calculus problem involving the change in water level of a half-cylinder-shaped water trough. I've worked through and understood most of the solution (I was unable to solve it on ...
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4answers
427 views

How is implicit differentiation formally defined?

I get that differentiation is an operation used on a function, so if a function is defined $x\mapsto x^2$, the derivative is $$ (x\mapsto x^2)' = x \mapsto \lim_{h\to 0} \frac{x^2+2xh+h^2-x^2}{h} ...
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4answers
45 views

How to calculate this integral involving an exponential?

I would like to calculate the integral $$\int_{0}^{\infty} xe^{-x(y+1)}dy.$$ I think I get the first steps correct. First $$\int_{0}^{\infty} xe^{-x(y+1)}dy = x\int_{0}^{\infty} e^{-x(y+1)}dy.$$ I ...
2
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1answer
43 views

prove $f(x)=\sum_{k=1}^{+\infty}\frac{1}{n}\cos^{n}x\sin(nx)$ is $\mathcal{C}^{1}(\mathbb{R}-\pi\mathbb{Z})$

let $f$ be a real valued function of a real variable defined by: $$f(x)=\sum_{k=1}^{+\infty}\frac{1}{n}\cos^{n}x\sin(nx)$$ Prove that $f(x)$ is $\displaystyle ...
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4answers
68 views

Derivative of $4x^{5x}$

I'm studying for an exam and I'm confused on these type of derivative problems. I know the answer I'm just confused as how to get to the answer. Would anyone mind going through the steps: Question: ...
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0answers
24 views

Jacobians and their Vector Coordinates

Define $f_2$ : $ℝ$ $\to$ $ℝ^2$ by putting $$f_2 (\theta)=(\cos(\theta),\sin(\theta)),$$ and for n $\ge3$ define $f_n: ℝ^{n-1}\toℝ^n$ inductively by setting $$f_n=(\theta_1, \theta_2, ...
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1answer
41 views

Differentiability at an end point of an open interval.

Suppose that $f : [a,b] \rightarrow \mathbb R$ is continuous on $[a,b]$, differentiable on $(a,b)$, and that $\lim_{x\rightarrow a^+}f'(x)=L$. Show that $f$ is differentiable at $a$, and that $f'(a) = ...
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3answers
332 views

Calculus: Maximum and Minimum Values

After the consumption of an alcoholic beverage, the concentration of alcohol in the bloodstream (blood alcohol concentration, or BAC) surges as the alcohol is absorbed, followed by a gradual decline ...
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1answer
12 views

Differentiating with fractions

Would somebody be able to tell me how to differentiate $2x^2(\frac {300-4x}{6x})$ ? Mainly, how would I be able to get the fraction into the normal $ax^2+bx+c$ form to differentiate? Thank you!
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1answer
35 views

Why is $\frac{d f(g(h(x)))}{d x} = \frac{d f(g(h(x)))}{d h(x)}\frac{d h(x)}{d x}$, not $\frac{d f(g(h(x)))}{d g(h(x)))}\frac{d h(x)}{d x}$?

According to the chain rule $(f(g(x))' = f'(g(x))g'(x)$ I would have thought the latter would be correct, since I would identify $g(x)$ with $g(h(x))$ in my question title.
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1answer
36 views

Prove that for any polynomial $P(x)= a_nx^n + \cdots +a_1x+a_0,P$ is differentiable

Prove that for any polynomial $P(x)= a_nx^n + \cdots +a_1x+a_0,P$ is differentiable, and $P'(x) = na_nx^{n-1}+\cdots+2a_2x+a_1.$ I am trying to figure out a way to prove this with out having to use ...
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4answers
102 views

Solving $\lim_{x\to-\infty}x^2\cdot e^x$ with L'Hopital

Use the L'Hopital rule to solve: $$\lim_{x\to-\infty}x^2\cdot e^x$$ I need a quotient of infinities or of zeroes. One way could be this: $$\frac{e^x}{x^{-2}} = \frac{0}{0}$$ So we apply ...
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3answers
46 views

Discuss the continuity and differentiability [closed]

Discuss the continuity and differentiability of $f$ at zero, where $$f(x) = \left\{ \begin{array}{ll} x^2 & \quad \text{if} \space x \gt 0 \\ 0 & \quad ...
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2answers
37 views

Derivative of $\ln(xy+1)=\sin(\pi x)$ at P(1,0) using implicit differentiation

Firstly, I confirmed P(1,0) is on the curve by substitution. Then I differentiated both sides giving me $\frac{x \frac {dy}{dx}+y}{xy+1}=\cos(\pi x)$ So $\frac {dy}{dx}=\frac {\cos(\pi ...
2
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1answer
45 views

How does this partial differentiation work?

Good day, There was a partial derivative in the lecture today that I can't comprehend. Can someone please explain how this works? Maybe there is a rule that I'm not thinking of right now. It's about ...
2
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1answer
61 views

Using chain rule to find the derivative of $(4x^2-2)^3$

I just took a quiz and one of the problems was to get the derivative of $f(x) = (4x^2-2)^3$. I used the chain rule and got $f'(x) = 24x(4x^2-2)^2$. However, plugging it into the derivative function ...
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1answer
18 views

Modification of derivation

Could you explain me how to modify: $$\frac{d}{dx}\left(e^{-7\sin(4x)\ln(x)}\right)$$ to this form: $$7x^{7sin(4x)} \cdot \left(4\cos(4x)\ln(x)+\frac{\sin(4x)}{x}\right).$$ Thank you :). I apologize ...
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0answers
10 views

Deriving a score test statistic to test value of the null hypothesis

http://i.stack.imgur.com/2fFJq.jpg When providing a specific estimator for ß, I suspect that we are using maximum likelihood to test for an estimator that has consistency, asymptotic normality, ...
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0answers
38 views

First-order derivative of a function with integration

Given this function $$f(x)=(\arctan(x))^2 \int_{\sqrt3}^{x^2}\frac{e^{-t}\sqrt t}{\ln(t^2+t)}dt$$ Calculate $f^{'}(\sqrt3)$
5
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1answer
109 views

Divergence in Definition of Laplace-Beltrami Operator

I am trying to derive an explicit formula for Laplace-Beltrami operator in global Cartesian coordinates for a special case of plane curve. I have found this article, and I would like to match their ...
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1answer
69 views

Let $R(s,t) = G(u(s,t), v(s,t))$, where $G$, $u$, and $v$ are differentiable. What is $R_s(1,2)$ and $R_t(1,2)$?

Here's everything that's given: $u(1,2)$ = $5$ $u_s(1,2)$ = $4$ $u_t(1,2)$ = $-3$ $v(1,2)$ = $7$ $v_s(1,2)$ = $2$ $v_t(1,2)$ = $6$ $G_u(5,7)$ = $9$ $G_v(5,7)$ = $-2$ I would post my attempt, ...
3
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1answer
45 views

If $g$ is differentiable and $g(1/n)=0$ for all $n$, then $g(0)=0$ and $g'(0)=0$

Suppose that $g:\mathbb{R}\rightarrow\mathbb{R}$ differentiable at $x=0$ and for each natural number $n$, $g(1/n)=0$. Prove that $g(0)=0$ and $g'(0)=0$ Since $g$ is differentiable at $x=0$, so ...
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1answer
32 views

Showing that given function is differentiable.

I am trying the following question from a competitive exam : I need to show that the following function is differentiable : f : $R$ --> $R$ defined as f(x) = $(1-x^2)$^$(3/2)$ if x is in (-1,1) and ...
2
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1answer
33 views

Term by Term Differentiability in the context of Uniform Convergence

I'm not sure how differentiability works with uniform convergence. My book says that we can show this (calculation wise) $$\varepsilon (x,a) = \sum_{k=1}^{\infty} E_{k}(x,a)$$ for some $x$ and $a$. ...
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1answer
26 views

If $f(x) = x^x\ln(5x-5)$, and $f'(x) = x^x\ln(5x-5)(g(x))$, Then What Would $g(x)$ Equal?

If $f(x) = x^x\ln(5x-5)$, and $f'(x) = x^x\ln(5x-5)(g(x))$, then what would $g(x)$ equal? So far, I have found the derivative: Find the derivative using the product rule, which gives ...
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0answers
178 views

How to Use Differentials to Estimate the Percentage Change in $r$, if $x$ increases by 6%. Let $r=6x^{-1/6}, x>0$

I am trying to determine how to use differentials to estimate the percentage change in $r$, if $x$ increases by 6%. Let $r=6x^{-1/6}, x>0$. So far, I have done the following steps: 1) Determine ...
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1answer
19 views

converge of ODE solution

for x(t) solution of $\dot{x}=f(x)$, f(x) differentiable and the derivative continuous. show that $lim_{t\to{\infty}}x(t)=+-\infty $ or $lim_{t\to{\infty}}x(t)=$stationary point it can ...
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1answer
18 views

Related rates: derivative of the function $A = \frac{x\cdot y}{2}$

Doing a related rates exercise, suppose you have this area formula for a triangle: $$A = \frac{x \cdot y}{2}$$ Where $A$, $x$ and $y$ are all functions with respect to time. I have to calculate the ...
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1answer
41 views

How to Find the Differential of $y=2\sin^2(x)$ when $x = \pi/4$ and $dx = 0.49$

I am wondering how to find the differential of $y=2\sin^2(x)$ when $x = \pi/4$ and $dx = 0.49$. I realize that I should be finding the derivative of $y=2\sin^2(x)$, which is $4\sin(x)\cos(x)$. And, I ...
0
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2answers
75 views

Finding the second derivative by differentiation

Given $x^5y + x +y^3 =3$, I have found the first derivative to be $x^5\, \frac{dy}{dx} + 5x^4y+1+3y^2 \, \frac{dy }{dx}= 0$. Need help calculating the second derivative.
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2answers
22 views

Help understand related rates problem: calculating the derivative of the distance function

I am having a particular problem understanding related rates problems. I think this is a good example to show my two issues: Two planes are flying towards the same point. The first is going to ...
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1answer
37 views

How to Find the Linear Approximation of $\ln(8-4x)$ at $x = 7/4$, and Use it to Estimate $ln(0.99)$

I am trying to determine how to find the linear approximation of $\ln(8-4x)$ at $x = 7/4$, and use it to estimate $\ln(0.99)$. So far, I have made the following steps: 1) Find the derivative of ...
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0answers
46 views

Find the interval in which g(x) is increasing and decreasing

Let $f'(\sin x)<0$ and $f"(\sin x)>0$ for all $x \in(0,\pi/2)$ and $g(x)=f(\sin x)+f(\cos x)$,then find the interval in which $g(x)$ is increasing and decreasing. In the second derivative of ...
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0answers
26 views

Math Analysis question about differentiation.

Suppose that $f$ is differentiable on $(a,+\infty)$. Show that if $f'(x)\to L$ as $x\to+\infty$, where $-\infty\le L\le +\infty$, then $\frac{f(x)}x\to L$ as $x\to +\infty$. Deduce that if $f(x)\to M$ ...
0
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1answer
82 views

First and Second derivatives differentiation

The equation $x^5y + x +y^3 =3$ defines implicitly a function $y=g(x)$ near $x=1$. Compute $g(1)$ , $g'(1)$, and $g''(1)$. If someone could show me the first few steps that would help.
2
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3answers
80 views

Can someone explain the integration of $\sqrt{v²+\tfrac14}$ to me?

I am currently trying to integrate this root: $$\sqrt{v^2+\frac{1}{4}}$$ According to several integration calculators on the web it is: $$\frac{\operatorname{arsinh}(2v)}{8} ...