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

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
43 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 ...
2
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2answers
111 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
54 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) ...
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1answer
30 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
40 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
70 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
23 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
43 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 ...
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1answer
37 views

Critical point of $F(x) = \int_0^{\pi x} \frac{d \theta}{ a \cos \theta - b \sin \theta}$

Let $a,b \in R$ such that $a>2b>0$ and let $F:[0, \frac{\pi}{3}] \rightarrow R$ be defined by $$F(x) = \int_0^{\pi x} \frac{d \theta}{ a \cos \theta - b \sin \theta}$$ How can one find a ...
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2answers
36 views

Finding Derivative using Limit Laws

Find the derivative of $\frac{x}{(1+2x)}$ using limit laws. I get stuck with the algebra once I set it up to $$\frac{\frac{x+h}{1+2x+2h} - \frac{x}{1+2x}}{h}$$
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1answer
27 views

Average rate of change over the given interval?

Find the average rate of change for the following functions please. I'm facing problems in these. $s=2t^3-5t+7$ interval from $t=1$ to $t=3$ $h=\sqrt{2t}-7$ interval from $t=8$ to $t=8.5$
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1answer
50 views

How to get $a,b$ and $c$?

I'm doing some exercises about calculus, and the exercise asked me to discover the values of $a$, $b$ and $c$. Be the function $y = x^2 + ax + b$ and the function $y = cx - x^2$ have the same tangent ...
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6answers
139 views

How to proof the the derivative of $x^2$ is $2x$?

I need your help, my question is how to proof that the derivative of $x^2$ is $2x$? Please I need a clear explanation.
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1answer
61 views

A calculus problem

Question: Suppose that $u(x,t)$ is continuous, together with its first and second partial derivatives; suppose that $u$ and its first partial derivatives are periodic in $x$ of period $1,$ and ...
1
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1answer
36 views

Decreasing derivative, so which is larger?

Suppose $f'(x)$ is a differentiable decreasing function for all $x$. In each of the following pairs, which number is the larger? Give a reason for your answer. A) $f'(5)$ and $f'(6)$ My answer is ...
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2answers
25 views

Calculus, local linearization

(a)Given that$$ f(7)=13$$ and$$ f′(7)=−0.38$$, estimate f(7.1). My answer was$$ f(6.1)= 13+ -0.38(x-7)$$ = 13.342. (b)Suppose also $$ f′′(x)<0 $$for all $x$. Does this make your answer to part ...
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2answers
33 views

If $y=2\sin^{-1}\sqrt{1-x}+\sin^{-1}(2\sqrt{x(1-x)})$ for $0<x<\displaystyle\frac{1}{2}$ then what is the value of $\displaystyle\frac{dy}{dx}$

If $y=2\sin^{-1}\sqrt{1-x}+\sin^{-1}(2\sqrt{x(1-x)})$ for $0<x<\displaystyle\frac{1}{2}$ then $\displaystyle\frac{dy}{dx}$ equals : A) $\displaystyle\frac{2}{\sqrt{x(1-x)}}$ B) ...
0
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1answer
47 views

$F(x) = \int_a^b \frac{x^2}{1+2\sin^3(t) + \sin^6(t) } dt$

Let $F(x) = \int_a^b \frac{x^2}{1+2\sin^3(t) + \sin^6(t) } dt$ i have to calculate the derivative of $F(x)$ with respect to $x$. Let $g(x) = \frac{x^2}{1+2\sin^3(x) + \sin^6(x)}$ then $g$ is ...
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2answers
54 views

Derivative of 1/sin x

I want to find out the derivative of 1/sin(x) without using the reciprocal rule. Let f(x) = 1/sin(x) Df/dx = (f(x+h) - f(x))/ h I keep getting 0 as the answer while the actual derivative according ...
1
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1answer
27 views

Finding $J'(0)$ and $J''(0)$?

The Bessel function of order $0$, $y= J(x)$, satisfies $$0 = xy'' + y' + xy$$ for all values of $x$. $J(0) = 1$. a) Find $J'(0)$. b) Find $J''(0)$. SOLUTION: a) $J'(0) = 0$. b) $J''(0) = -1/2$ ...
11
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1answer
104 views

Differentiation of a function $f:\mathbb{Q}\to \mathbb{Q}$(Rational Calculus)

Assume that $f:\mathbb{Q}\to \mathbb{Q}$ is given such that $\forall a\in \mathbb{Q}$ the following limit, exists \begin{equation} \lim_{x\to a} \frac{f(x)-f(a)}{x-a}\in \mathbb{R} ...
0
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1answer
38 views

I need help on deriving this trig function.

I am trying to derive $e^x \sin x - 2x \csc x$. I tried using the product and difference rule. So I got the derivative for $e^x \sin x$ and got $(e^x)(\cos(x))+(\sin(x))(e^x)$ and for the derivative ...
0
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2answers
42 views

$f : \mathbb R^n \to \mathbb R$, what is the gradient of $f(tx)$?

Fairly simple question, suppose there is a function $f: \mathbb R^n \to \mathbb R$, and a scalar $t \in \mathbb R$. is it possible to find $D_f(tx)$ using only $t$ and $D_f(x)$? Perhaps using chain ...
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0answers
91 views

Will antiderivative always be differentiable?

Suppose f(x) is continuous on [0,1]. Obviously, such a function will be integrable. Will antiderivative be always differentiable on (0,1)? The answer is "Yes" by the Fundamental Theorem of Calculus. ...
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3answers
87 views

Derivative in calculus $f(t)= 7\sinh(\ln t)$

How to find the derivative of this function $$ 7\sinh(\ln t)?$$ I don't know from where to start, so i looked at it in wolfram alpha and it was saying that the $$ 7((-1 + t^2) / 2t) $$ I did not get ...
0
votes
1answer
39 views

Finding Derivative of a complicated function

I have the following function I want to be able to feed into the Newton-Rhapson root solving algorithm (C++ boost), since an analytical solution is not possible:$$f(p, k, n, Pr)=\left(\sum_{i = ...
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1answer
23 views

Find all points where $f$ is differetiable

Let $$f(x) = \left\{ {\matrix{ {0,x \notin Q} \cr {{x^2}({x^2} - 1),x \in Q} \cr } } \right.$$ I already proofed that $f(x_0)$ is continuous iff $x_0\in \left\{0,1,-1\right\}$. Now, if $f$ ...
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1answer
56 views

Differentiation of multivariable function proof

I'm looking for the differentiation of multivariable function integral $$\frac{\mathrm{d} }{\mathrm{d} x} \int_{v(x)}^{u(x)}f(t,x)dt=u'(x)f(u(x),x)-v'(x)f(v(x),x)+\int_{v(x)}^{u(x)}\frac{\partial ...
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1answer
20 views

Finding all continuity and differentiability points of a function

Let $$f(x) = \begin{cases} x^2(x^2-1),&x \in\mathbb{Q} \\ 0,&x \not\in\mathbb{Q} \end{cases}$$ A. When is this function continuous? when is it differentiable? I solved these kind of ...
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3answers
53 views

How do I simplify this derivative?

I'm trying to find the derivative of $$f(x)=\frac{4+(1/x)}{(x+4)}$$ I applied the quotient rule and I got as far as $(-4-(2/x)-(4/x^2))/(x+4)^2$. The final answer is $(-4x^2+2x+4)/(x^2(x+4)^2))$ ...
1
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1answer
20 views

Optimization of parallelepiped.

Let $K \in R^3$ the ellipsoid given by the equation $ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $ with $a,b,c > 0$ , let $(x,y,z) \in K$ on the first octant, consider the ...
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6answers
133 views

Derivatives of equations

Assume that $x$ and $y$ are related by the equation $y\ln x=e^{1−x}+y^3$. Compute $dy/dx$ evaluated at $x=1$. I do not understand how to compute the derivative of an equation. Please explain.
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1answer
33 views

$f$ a differentiable fucntion in $[a,b]$ with $f´(a) < C < f´(b)$

Let $f$ a differentiable fucntion in $[a,b]$, suppose the existence of a point $C$ with $f´(a) < C < f´(b)$ how can i deduce that given the function $g(x) = f(x) - C(x-a)$ then exist a pint ...
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1answer
56 views

Calculating $\frac{d}{d(x^2)}f(x)$

There's a question I need to solve, which requires that I take the derivative of some function by the square of a variable, and I'm not sure how to do such a thing. For example: $\frac{dx}{d(x^2)}$ - ...
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2answers
44 views

Prove an inequality (Using Taylor expansion)

Prove: $\frac{x}{1+x} < \ln(1+x) < x$. I thought a good practice would be to prove it using Taylor Expansion. Here's my try: $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}...$$ The n=1 ...
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1answer
30 views

How can I calculate the derivative of a Catmull-Rom spline with nonuniform parameterization?

Allow me to preface this by saying I am not a trained mathematician in any sense, so it's entirely possible I'm missing something rather fundamental. That said, I'm trying to take the derivative of a ...
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0answers
18 views

Prove or disprove the statement related to the definition of multivariable-differentiable function

The question: Let $f,f_1,...,f_n \; (n > 0)$ be functions from $\mathrm{D} \subset\mathbb{R}^n$ to $\mathbb{R}$ satisfying $$\left ( \sqrt{\sum_{i=1}^n x_i^2} \right ) f(\mathrm{x}) = \sum_{i=1}^n ...
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3answers
40 views

Derivative of sum of two exponential functions

I have the following formula - $$ f(x) = \left(0.1 e^{-1.5{x}^{0.2}} + 0.9 e^{-0.5{x}^{0.1}}\right)^{c}$$ where $\bf c$ is a constant value. How can I solve $f'(x)$ ? According to the answer, I ...
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1answer
19 views

Maximize area of a rectangle between parabola and a line

I was given a task to maximize the area of a rectangle that can be inscribed between parabola $y=1-x^2$ and a line $y=0$ such that one side of the rectangle lies on the $x$ axis. My idea is to somehow ...
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1answer
37 views

Why is $(\sec x)' = \tan x\sec x$ and not $\tan x$?

As far as I understood, the Fundamental Theorem of Calculus states that the integral of a function is its anti-derivative. And yet, although the integral of $\tan x$ is $\sec x$, the derivative of ...
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4answers
40 views

derivative $\frac{df(t)}{dt}$ of $f(t) = \int_0^t\ln{(s^2+t^2)} ds$

Let $f(t) = \int_0^t \ln{(s^2+t^2)} ds$, how can I find the derivative $\frac{df(t)}{dt}$? The function $\,\int_0^t \ln{(s^2+t^2)} ds$ is defined to be continuous in $s^2+t^2 > 0$ and $ s^2+t^2 ...
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4answers
80 views

Local minimum of $f(x) = 4x + \frac{9\pi^2}{x} + \sin x$

What's the minimum value of the function $$f(x) = 4x + \frac{9\pi^2}{x} + \sin x$$ for $0 < x < +\infty$? The answer should be $12\pi - 1$, but I get stuck with the expression involving both ...
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0answers
29 views

Last step in derivation of Euler-Lagrange equation (definite integral)

In the classical derivation of Euler-Lagrange equation in the calculus of variations, for a case with fixed end points at $x=a$ and $x=b$, we have the final step in derivation arriving at: ...
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1answer
108 views

showing $\int _a^b\left(f'\left(x\right)\right)dx\:=\:f\left(b\right)-f\left(a\right)$

Let $f(x):[a,b]\to \mathbb R$, be differentiable on $[a,b]$ (and continuous) so that $f'(x)$ is integrable on $[a,b]$. I need to show that: $$\int _a^b\left(f'\left(x\right)\right)\mathrm dx = ...
0
votes
1answer
20 views

Direction for greatest derivative

Suppose I have a function like $f(x,y) = e^x e^y x^2 y^2$, and I want to know in which direction the derivative will grow fastest at a stationary point. $(0,0)$ is a stationary point of the example ...
0
votes
1answer
30 views

Finding tangent line of $f(x) = 1/x$

Find the equation of the tangent line to $f(x) = \dfrac{1}{x}$ through the point $(0, \alpha)$. Answer: $y = −\alpha^2\dfrac{x}{4} + \alpha$ I've found $f'(x)=-\dfrac{1}{x^2}$ but how do I find ...
1
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2answers
46 views

Derivative function went wrong

I am trying to take the derivative of this function but I am facing some difficulties. $$f(x)= e^{\ln(e^{7x^2+11})}$$ My answer was : $7e^{(7(x^2))}*14x$ I cancelled the $\ln$ with the $e$ first, ...
2
votes
2answers
68 views

Prove $\sqrt{x}>\ln(x)$ in $[1,\infty)$

Well, i try to prove this statement. i choose to make function: $f\left(x\right)\:=\:\sqrt{x}-\ln x$ but the derivative is: $\dfrac{\sqrt{x}\:-\:2}{2\sqrt{x}}$ and it's not always greater than $ 0$. ...
5
votes
0answers
76 views

general solution of the equation $\frac{dy}{dx} =\exp(y/x)$

How can i get the general solution of the equation a) $\frac{dy}{dx} = \exp(y/x)$ b) $\frac{dy}{dx} = \exp(x-y)$ and $y=2$ when $x = 0$ I tried b) first: This is a first-order nonlinear ordinary ...