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

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3answers
48 views

How to differentiate $-x^3(3x^4-2)$

What am I doing wrong? $-x^3*d/dx(3x^4-2)+(3x^4-2)*d/dx(-x^3)$ $-x^3(12x^3-2)+(3x^4-2)(-3x^2)$ $-12x^9+2x^3-9x^6+6x^2$ When just using the power rule it comes out to be $-21x^6+6x^2$
0
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2answers
28 views

Is there an easy way to study the sign of this?

I would like to study the sign of this derivate, but I don't know where to start : http://www.wolframalpha.com/input/?i=derivate+sqrt%28x%5E4-7x%5E2%2B16%29
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2answers
41 views

Why cant we do substitution in differentiation but is ok in taylor series?

I have the same question 10 year ago when i was studying high school. I dont understand it and i give up the math. 10 year ago, i need to work with calculus during work and this question come to find ...
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1answer
27 views

demonstrate that function is increasing in intervals that are multiples of pi?

I have the derivative: $$- \frac1{x^2} + 1 + \frac{\cos^2(x)}{\sin^2(x)}$$ and am supposed to show that this is positive for all $x \in (n\pi, (n+1)\pi)$. How exactly am I supposed to do that? ...
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3answers
82 views

Am I differentiating this wrong?

Differentiation is the opposite of Integration $$\begin{align}\int \cos^2x dx\end{align}$$ $$\begin{align}-\frac{\cos^3x}{3\sin x}\end{align}$$ Now if we differentiate $-\frac{\cos^3x}{3\sin x}$ we ...
0
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1answer
69 views

Prove $\sin(x)< x$ when $x>0$ using LMVT

According to Lagrange's Mean Value Theorem (LMVT), if a function $f(x)$ is continuous on $\left[a,b\right]$ and differentiable on $\left(a,b\right)$, then there exists some constant $c$ such that ...
1
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1answer
40 views

What to do *rigorously* when the second derivative test is inconclusive?

How do you rigorously check if a point is a local minimum when the second derivative test is inconclusive? Does there exist a way to do this in general for arbitrary smooth (or analytic...) functions? ...
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1answer
30 views

Derivative of $f(x)=\frac{7x^3+3x+30}{\sqrt{x}}$

$$f(x)=\frac{7x^3+3x+30}{\sqrt{x}}$$ $f^{\prime}(x)=\dfrac{\dfrac{1}{2\sqrt{x}}(7x^3+3x+30)-(21x^2+3)(\sqrt{x})}{(x^{1/2})^2}$ ...
2
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0answers
11 views

Proof that maximal interval of existence exist and bounded

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
1
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1answer
30 views

Derivative of $f(u)=\sqrt{8} \;u+\sqrt{6u}$

$$f(u)=\sqrt{8} \;u+\sqrt{6u}$$ $f(u)=\sqrt{8}\;u+(6u)^{1/2}$ $f^{\prime}(u)=\sqrt{8}+\dfrac{1}{2}(6u)^{-1/2}$ $=\sqrt{8}+3u^{-1/2}$ This was marked wrong, though. What am I doing wrong? ...
1
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1answer
21 views

If $A=5r^2$ and $\frac{dA}{du}=2$, what is $\frac{dr}{du}$

I am unsure exactly what this question is asking me to do. I think $\frac{dA}{dr} = 10r$ and I assume $u=a/2$ but I'm not sure where to go from there.
2
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1answer
30 views

Jacobian matrix of the inverse of a bijective function

Let $f:\mathbb{C}^n\rightarrow\mathbb{C}^n$ be a function such that $f=f(f_1,\ldots,f_n)$ and $f_i=f_i(x_1,\ldots,x_n)$. Also, $f$ is bijective and its Jacobian matrix exists. Does$f^{-1}\,$Jacobian ...
-5
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0answers
24 views

Change order between integral and differential calculation

Are those right? And I want to ask, in general case, when we can change the order of diff and integral: diff(integrate(L(x,y))) integrate(diff(L(x,y)))
0
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1answer
26 views

If $f$ is $C^1(\mathbb{R})$, is it $C^1(\{a\})$?

Say I have a well-behaved function like $f(x)=x$. This is obviously $C^1$, but does it make sense to say the function is $C^1$ around a single point? A broader question, if $a\in\mathbb{R}$, does ...
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1answer
176 views
+300

Existence of a bounded function satisfying a second order differential equation

This question is a variation version from here. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ ...
0
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3answers
63 views

Is it even possible to find the variance of this moment generating function?

This is my moment generating function: $M_x(t) = \frac{6e^t}{t^2} + \frac{6}{t^2} + \frac{12e^t}{t} - \frac{12e^t}{t^3} + \frac{12}{t^3}$. I have to find the mean the variance of it. After taking ...
3
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1answer
59 views

Suppose all partial derivatives of $f$ exist at $x_0$; is $f$ continuous at $x_0$?

Consider $f : C \to \mathbb{R}$ with $C \subset \mathbb{R}^n$ being open: Suppose $f$ is differentiable at $\mathbf{x}_0 \in C$. Is $f$ continuous at $\mathbf{x}_0$? Why? Suppose all partial ...
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0answers
13 views

Calculus Single Variable: Find max and min of hard to graph function

Consider the function F defined by F(x)= integral from 0 to x of $t|sint(t)|dt$. Find the absolute maximum value and absolute minimum value of y=f(x). I know there's one at x= zero but the ones ...
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1answer
23 views

Evaluation of derivative: if $p(x)=b_0 + (x-z)q(x)$, then $p'(z)=q(z)$

I just wanted to confirm that I did this correctly, because this answer seemed too easy to obtain: $p(x)=b_0 + (x-z)(q(x)).$ Show that $p'(z)=q(z).$ My answer: $$\begin{align*} p'(x) ...
3
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3answers
51 views

If $f$ satisfies $\forall x\in\Bbb{R},0\leq f'(x), f''(x)$ and if $\exists a\in\Bbb{R}$ such that $0<f'(a)$, Then $lim_{x\to\infty}f(x)=\infty$

I got this problem: Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function that satisfy $\forall x\in\mathbb{R},0\leq f'(x)$ and $0\leq f''(x)$ Prove that if $\exists a\in\mathbb{R}$ ...
3
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4answers
72 views

Existence of solution in $x,y \in (a,b)$ of $ \bigg(\dfrac { a+b}2\bigg)^{x+y}=a^xb^y$

Let $a<b$ be positive real numbers , then is it true that there exist $x,y \in (a,b)$ such that $ \bigg(\dfrac { a+b}2\bigg)^{x+y}=a^xb^y$ ?
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1answer
21 views

Calculating the directional derivative of $ xy^3/(x^3+y^6)$.

$f(x,y)= xy^3/(x^3+y^6)$ if $(x,y)\neq 0$, $f(x,y)=0$ if $(x,y)=0$. Prove that $f'(0; a)$ exists for every vector $a$. I know how to find the directional derivative from limit equation, but don't ...
0
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1answer
42 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
29 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 ...
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0answers
30 views

Proving that maximal interval of existence exists and that solution is unque

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
1
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2answers
83 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 ...
0
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1answer
59 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 $$ \lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2}=0\,\,\,\text{for all }x, $$ then ...
1
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2answers
36 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} ...
1
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1answer
54 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
14 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 ...
0
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0answers
30 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 $f'(z)=u_x+iv_x$.
0
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2answers
25 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
54 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
64 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 ...
0
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0answers
29 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
59 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
51 views

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

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
59 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
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 ...
0
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4answers
31 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
32 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 ...
1
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2answers
34 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 ...
1
vote
2answers
55 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
40 views

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

Here $f$ and $r$ are constants. $$e(x) = \frac {x^2 + 80x + 40f}{rx}$$
1
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1answer
78 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
58 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 ...
8
votes
1answer
202 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 ? What ...
0
votes
1answer
14 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
votes
2answers
25 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) ...
0
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1answer
136 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 ...