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

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49 views

I'm having trouble with this question on derivatives.

Carlos is blowing air into a spherical soap bubble at the rate of $7 \mathrm{cm}^3/ \mathrm{sec}$. How fast is the radius of the bubble changing when the radius is $11 \mathrm{cm}$? (Round your answer ...
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1answer
62 views

Find lowest and highest value of function $f(x)=\int_0^x{\frac{2t-2}{t^2-2t+2}}dt$

Find highest and lowest value of function: $$f(x)=\int_0^x{\frac{2t-2}{t^2-2t+2}}dt$$ We need to use first derivative test to find critical points. $$f'(x) = \frac{2x-2}{x^2-2x+2}(x)' - ...
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2answers
59 views

How do I go about solving this derivative of inverse tangent?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$8\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=8\frac{1}{1+x^2}$$would ...
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3answers
100 views

Show that it is possible that the limit $\displaystyle{\lim_{x \rightarrow +\infty} f'(x)} $ does not exist.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ a differentiable function with continuous derivative and the limit $\displaystyle{\lim_{x \rightarrow +\infty} f(x) }$ exists. Show with an example that it ...
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3answers
170 views

A counter-example to differential function but not twice differential

Find a function $f$ that is differentiable, but not twice differentiable and which does not belong to the following type: $$f(x) = \begin{cases} x^\alpha \sin(x^{\beta}) & x \neq 0 \\ 0 & ...
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1answer
34 views

How do I solve this trig derivative in respect to $x$?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=\frac{1}{1+x^2}$$would ...
2
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1answer
84 views

Characterization of differentiable functions from $\mathbb{R}^m$ to $\mathbb{R}^n$.

Let $U\subset\mathbb{R}^m$ be an open set. Consider a function $f:U\to\mathbb{R}^n$ and a point $a\in U$. I need help to prove that the following sentences are equivalents. (a) There exists a ...
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3answers
53 views

I need help finding the derivative of this natural logarithm function.

Okay so $$f(x)=\ln[x\ln(x+2)]$$ so $$\ln(x)+\ln(\ln(x+2))$$so $$1.a\frac{dy}{dx}\ln(x)=\frac{1}{x}$$and I thought by chain rule that ...
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2answers
265 views

I am having problems figuring out how to derive this.

I have the function $$\tag{1} f(x)=\ln\sqrt{8+\cos^2x}$$ So we derive it as follows: $$\tag{2} f(x)=\ln(8+\cos^2x)^\frac{1}{2}$$ $$\tag{3} f(x)=\frac{1}{2}\ln(8+\cos^2x)$$ $$\tag{4} ...
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90 views

Find $G'\left( x\right)$.

Let $$G\left( x\right)=\int_{x}^{2x}{f\left( t\right)dt}$$ Find $G'\left( x\right).$ I tried to divide the integration interval but the subintervals are expressed in terms of $x$.
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2answers
101 views

How do I go about solving this derivative?

I have the function $$f(x)=\ln\sqrt{8+\cos^2x}$$ so $$1.f(x)=\ln(8+\cos^2x)^\frac{1}{2}$$so$$2.f(x)=\frac{1}{2}\ln(8+\cos^2x)$$so $$3.f'(x)=\frac{1}{2}\left[\frac{-2 \cos x^{\sin x}}{8+\cos ...
2
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2answers
57 views

I need help on the process of solving this derivative.

How do I go about solving this derivative. $$f(x)=\ln\left(\frac{7x}{x+4}\right)$$ I go from this to $$1. \quad f(x)=\ln(7)+\ln(x)-\ln(x+4)$$ and then $$2. \quad f'(x)=\frac{1}{x}-\frac{1}{x+4}$$ then ...
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1answer
133 views

relations between differential, partial derivative, directional derivative

I am a bit lost. Could you explain me relations between differential, partial derivative, directional derivative? I mean that I need some theorem and proofs that for example if differential exists ...
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2answers
78 views

I need help finding the derivative of the inverse function.

So $$f(x)=\frac{x+1}{2x-1}$$ and $$g(x)$$ is an inverse of $$f(x)$$ I have the points on $f(x)$ of (2,1). So I know that $f(2)=1$, $g(1)=2$ and $g'(1)=\frac{1}{f'[g(1)]}$ so $g'(1)=\frac{1}{f'(2)}$ ...
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3answers
88 views

Simple Differentiation Problem Involving Area Radius and Circumference

A stone is dropped into a pool of water, and the area covered by the spreading ripple increases at a rate of $4 m^2 s^{-1} $. Calculate the rate at which the circumference of the circle formed is ...
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4answers
141 views

Why is the derivative at a jump undefined even if the slope remains the same?

I've searched online and found almost nothing. What in the mathematical definition of a derivative makes it so that the derivative of the following is undefined at ...
2
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1answer
74 views

Figuring out when $f(x) = \sin(x^2)$ is increasing and decreasing

Regarding the function $f(x) = \sin(x^2)$, I'm supposed to figure out when it is increasing/decreasing. So far, I've found the derivative to be $f'(x) = 2x\cos(x^2)$. So long as I can solve the ...
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3answers
78 views

Calculate value of $ f'(0)$ of function $f(x) = \sum_{n=1}^{\infty} \frac {\sin(nx)}{n^3}, f: {\bf R}\to{\bf R}.$

I tried to solve it in the following way: $$f'(x) = \left(\sum_{n=1}^\infty \frac{\sin(nx)}{n^3}\right)'= \sum_{n=1}^\infty \left( \frac{\sin(nx)}{n^3}\right)'= \sum_{n=1}^\infty \frac{(n x \cos(n ...
2
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1answer
203 views

Differentiation of a parametric function using MATLAB or Maple

I'm doing some mathematical calculation of some symbolic math that includes multiplication and differentiation of some matrices. Some of the parameters in my calculations are functions of time. for ...
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2answers
97 views

How to solve complex derivatives with multiple terms?

I have several derivatives to find: For $g(s)=3s^3-s+4$, $g'(s)=$ For $p(t)=\frac1t+t^2$, $\frac{\mathrm dp}{\mathrm dt}=$ For $w(u)=\sqrt u-2u^2-10$, $\frac{\mathrm dw}{\mathrm du}=$ ...
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4answers
6k views

How to find the derivative of a fraction?

So I have this fancy problem I've been working on for two days: I need to find two things: 1) $f'(t)$ 2) $f'(2)$ I have tried plugging it into the definition of a derivative, but do not know how to ...
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1answer
55 views

Differentiation involving determinant

This question has arisen by following the proof in the appendix of Louis Liporace's paper on maximum-likelihood estimation, where the paper concerns classes of probabilistic functions (elliptically ...
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1answer
64 views

Values of $6 + \int_a^x \frac{f(t)}{t^2} dt = 2 \sqrt{x}$

Let $f$ and $a$ such that $6 + \int_a^x \frac{f(t)}{t^2} dt = 2 \sqrt{x}$. I need to find the values of $f$ and $a$ that satisfies this condition. For this i tried: $F(x) = 6 + \int_a^x ...
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2answers
64 views

why is the chain rule used for the area function $A=\frac{1}{2}xy$

To differentiate the area of a triangle function, $A=\dfrac{1}{2}xy$ with respect to time $t$, my text says to use the chain rule and the product rule. So it would be: ...
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3answers
580 views

How to calculate a derivative using the “Power Rule” If it includes a negative exponent?

So my understanding of the power rule is that you take your problem with an exponent like this: $x^5 = 5x^4$ or for $x^n$, $f'(x)=nx^{n-1}$ However, it does not seem to be working for me when ...
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2answers
76 views

What am I doing wrong with this derivative? (Calculus)

I've been doing derivatives with the formula: Definition of a Derivative: for every $x$ plugin $(x+h)$, then subtract original from the equation. This means for $x^2$, I get: $$\frac{(x+h)^2 - ...
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1answer
35 views

critical points - calculus

$$f(x) = 5x^{1/5} − x $$ $$f^\prime (x)= x^{-4/5} -1 $$ The question is: Find the $x$ values of all of the critical points. Enter your answers as a comma separated list. As far as I know that ...
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2answers
96 views

L'Hôpital's Rule and Infinite Limits

I was wondering if anyone could help me with computing a limit using L'Hôpital's Rule. Using L'Hôpital Rule for the following limit, I get the following result: \begin{equation} \lim_{x \to 0} ...
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2answers
44 views

Find the value of derivative, given that the tangent line passes through a particular point

If the line tangent to the graph of the function $f$ at the point $(2,7)$ passes through the point $(-3,-3)$ then $f'(2)$ is...? A. 5 B.1 C. 2 D.7 E. Undefined I don't understand how to do this. I ...
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1answer
37 views

Prove that $\arcsin (x)$ differentiates to $\frac{1}{ \sqrt {1-x^2}}$ [closed]

I want to prove this. I have no idea where to start. How do I do it?
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1answer
43 views

Optimal way to find derivative - numerically

Suppose we are given points $x_0,x_1,x_2$ evenly spaced points $(x_0-x_1=x_1-x_2)$, and $u(x_1),u(x_2),u(x_3)$ Where $u$ is some function. Find the best way to approximate $u''(x)$ using only the ...
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1answer
880 views

How to prove that Δy/Δx = f(x+Δx)-f(x) / Δx?

How do I prove that $$\frac{\Delta y}{\Delta x} = \frac{f(x+Δx)-f(x)} {Δx}.$$ I know that this is the slope formula to find the derivative of a function $y=f(x)$ and I know that the formula for a ...
2
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1answer
29 views

if $f([a,b])=[c,d]$ and $[c,d] \subset [a,b]$, is there $x \in [c,d]$ such that $f(x)=x$?

I'm trying to prove something that I'm not sure is correct. Let $f$ be a continuous, differentiable and monotonic function $f:[a,b] \to [c,d]$, where $[c,d] \subset [a,b]$. Is there an $x \in [c,d]$ ...
2
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2answers
153 views

Find the inflection points in the graph

The question is which of the $x$-values of the given points are inflection points of the function $f(x)$ itself? I chose $C,F$ and $H$ because at this point the $f'(x)$ is zero. But my answer was ...
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1answer
216 views

True/ False to strength my knowledge in calculus

Q1) For a continuous function $f$ whose domain is all real numbers, if $f'(p)$ is undefined, then $x = p$ could be a local maximum or minimum of $f$. I say that at corners it could be maxima or ...
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2answers
45 views

mean value property of derivatives in high dimensions

Let $E$ be a path-connected subset of $\mathbb{R}^n$ and $f$ a differentiable function on $E$. Prove or disprove: for any $x,y\in E$, there exists $z\in E$ such that $f(x)-f(y)=\nabla f(z)\cdot ...
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1answer
47 views

Fractional Derivatives on a function with bounded Support

I have a question about functions that have bounded support in $\mathbb{R}$. In particular, suppose that I have a function $f$ with support $A\subset \mathbb{R}$ so that $A$ is compact. Without loss ...
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4answers
38 views

Derivative of $(5x-2)^3$

How is the derivative of $(5x-2)^3$ equal to $15(5x-2)^2$ and not $3(5x-2)^2$. According to $\frac{df}{dx} = nx^{n-1}$, it has to be $3(5x-2)^2$ right. Please explain.
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1answer
43 views

Derivative of a Trigonometric Function Help

Trying to derive a trigonometric function, Wolfram Alpha and my textbook provide two different answers. Here is the function: $$y = {\cot x\over (1+\csc x)}$$ First step using quotient rule results ...
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1answer
40 views

Showing that $\frac{d^m}{dx^m}[f(x)g(x)] + {{m+1}\choose k}\frac{d^{m+1+k}}{dx^{m+1+k}}f(x)\frac{d^k}{dx^k}g(x) = \frac{d^{m+1}}{dx^{m+1}}[f(x)g(x)]$

I need to algebraically, or using basic calculus, show that $$\displaystyle\frac{d^m}{dx^m}[f(x)g(x)] + {{m+1}\choose k}\frac{d^{m+1+k}}{dx^{m+1+k}}f(x)\frac{d^k}{dx^k}g(x) = ...
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2answers
88 views

Help finding the second derivative of this function.

I need help finding the second derivative of this function. I found the first derivative and the second, but the program says my answer is incorrect either by typing error and it won't process ...
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1answer
36 views

Newton derivative of the distance function in $\Bbb R^2$

If we consider the distance function $d$, where $d(x)=dist(x,\partial\Omega)=\inf_{y\in\Omega}\|x-y\|_2$, how would one calculate the derivative in some direction $v$, i.e. ...
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1answer
45 views

Derivative of $\frac{1}{2s}-\frac{5}{4s^{3}}$

$r=\dfrac{1}{2s}-\dfrac{5}{4s^3}$ $r=\dfrac{1}{2}s^{-1}-\dfrac{5}{4}s^{-3}$ $r^{\prime}=-\dfrac{1}{2}s^{-2}-\dfrac{5}{4}(-3)s^{-4}$ $r^{\prime}=-\dfrac{1}{2s^{2}}+\dfrac{15}{4s^{4}}$ Is this ...
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2answers
123 views

$f\left(\frac{1}n\right)=\frac{n^2}{n^2+1}$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ of class $C^\infty$ $\forall n\in\mathbb{N}^*,f\left(\frac{1}n\right)=\frac{n^2}{n^2+1}$ Let $p\in\mathbb{N}^*$ What is the value of $f^{(p)}(0)$ ? (by ...
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2answers
67 views

It is impossible to put $x = f(x)g(x)$, where $f$ and $g$ are differentiable Functions and $f(0) = g(0) = 0$

I have a question that i don't understand and it was mentioned in my textbook of calculus. why It is impossible to put $x =f(x)g(x)$, where $f$ and $g$ are differentiable Functions and $f(0) = g(0) ...
0
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1answer
202 views

How do we know when two curves touch each other?

What are the conditions of two curves touching each other? A necessary condition for this is that the derivative for both the curves should be the same at the point of intersection. But that doesn't ...
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2answers
41 views

How do I go about answering this question on derivatives?

$$ \begin{align}Find&& f''(2)&&if&&f(x)=x^2f(2x) &&and \end{align}$$ $$ \begin{align} &f(4)=-2,f'(4)=1,f''(4)=-1 \end{align}$$ I have no idea on how to approach ...
2
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1answer
160 views

Checking derivation of y = a^x

Can you tell me if there are any flaws with this derivation of $y = a^x$... The assumptions are that the derivative $$\frac{d}{dx}e^x = e^x$$ and that the derivative $$\frac{d}{dx}\ln x = ...
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1answer
25 views

The number of points in the rectangle which lie on the curve $y^2=x+\sin x$ and at which the tangent to the curve is parallel to the $X-$axis

The number of points in the rectangle : $\{(x,y)|-10\le x\le10$ and $-3\le y\le3\}$ which lie on the curve $y^2=x+\sin x$ and at which the tangent to the curve is parallel to the $X-$axis, is A) $0$ ...
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2answers
97 views

Extreme value problem, maximize ratio of volume to surface area

For a cylindrical can, how to choose the ratio of the height to the radius such that the ratio of the volume to the surface area gets maximized? The volume ist $V = \pi r^2 h$ and the surface ...