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

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0
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2answers
28 views

Derivative of function defined by integral of different variable

I have the following exercise which I certainly have gotten no clue about it. Let F(t) be defined: $F(t) = \int_{tan(t)}^{\sqrt{t^2+1}} e^{-tx^2}dx$ What is $F'(0)$? I have no clue about ...
2
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0answers
20 views

Sign function identity proof

The signum function is defined by$$sgn(t)=\left\{\begin{matrix}-1, \ t<0\\0, \ t=0 \\ 1, \ t>0 \end{matrix}\right.$$has derivative$$\frac{d}{dt} sign(t) = 2 \delta(t)$$Use this result to show ...
1
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0answers
32 views

Specific fucnction has 11 different zeros

Let $f : \mathbb{C} \to \mathbb{C}$ be given by $$ f(z) = z^{11} + 4 e^{z + 1} - 2 $$ Show that $f$ has 11 different zeros in the annulus $\{z \in \mathbb{C} : 1 < |z| < 3\}$. This is an old ...
1
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3answers
34 views

Third derivative of $y=at^2+2bt+c$ and $t=ax^2+bx+c$

If $y=at^2+2bt+c$ and $t=ax^2+bx+c$. Then find $$\frac{d^3y}{dx^3}$$ Now $\frac{dy}{dx}=(2at+2b).(2ax+2b)$ but to proceed further as $\frac{dy}{dx}$ is function of $x,t$
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0answers
25 views

Successive differentiation

Find the value of $y_n$ for $x = 0 $,when $$ y = e^{(a sin^{-1}(x))}$$. In the my book its already solved the problem is that I don't understand after a certain point ,the steps. After solving we ...
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2answers
70 views

Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f(0)=0$ and $|f'(x)|\leq1 \forall x\in\mathbb R$

Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f(0)=0$ and $|f'(x)|\leq1 \forall x\in\mathbb R$. Then there exists $C$ in $\mathbb R $ such that $|f(x)|\leq C \sqrt |x|$ ...
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1answer
31 views

Double derivative w.r.t x and y needed

I have the following function $$h(x,y)=\int_{\frac{eaf}{c(1-x)}}^\infty e^{-t-\frac{eagf}{cx(1-y)t}}dt$$ where $a,c,e,f,g$ are constants. I need to find the double derivative w.r.t. $x$ and $y$ i.e. $...
38
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8answers
2k views

Does this pattern have anything to do with derivatives?

In 6th grade I was first introduced to the idea of a function in the form of tables. The input would be "n" and the output "$f_n$" would be some modification of the input. I remember finding a pattern ...
3
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2answers
35 views

Conjecture about Cal 1 derivatives?

Conjecture: Let $F\left(\vec{x}\right) : \Bbb{R}^n \to \Bbb{R}$ Define $g(t) = F(t, t, \dots, t)$ Then $$g^{\prime} (t) = \left(\sum_{i=1}^n \ { \partial F \over \partial x_i}\right)\...
1
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1answer
31 views

Repeating/“Periodic” Derivatives? [duplicate]

We know that $Ce^x$ and $0$ are the two functions whose first derivative is equal to itself, but what about derivatives of a higher order? For example, the second derivative of $e^{-x}$ is equal to ...
1
vote
1answer
23 views

What condition on the coefficients $a_n$ will guarantee $f(x) = \Sigma_{n = -\infty}^{\infty}a_{n}e^{2\pi inx}$ is k times differentiable?

$f(x) = \Sigma_{n = -\infty}^{\infty}a_{n}e^{2\pi inx}$ What condition on the coefficients $a_n$ will guarantee $f$ is $k$ times differentiable? I'm not sure where to begin with this, because it ...
3
votes
1answer
45 views

Derivate a function defined by an integral, whose variable are the integration limits

I have to find the derivative of the following one-variable function and evalue it for $t=0$: $$g(t)=\int_t^{t^2} \cos(tx)dx$$ In class, we saw a formula that says that a function such as $$F(t)=\...
5
votes
2answers
76 views

Pork roast defrost using calculus

I am really stuck on this problem for calculus and I could use some help A pork roast is removed from the freezer and left on the counter to defrost. The temperature of the pork roast was $−4^\circ C$...
0
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0answers
34 views

differentials in physics [migrated]

Often I find the following expressions in physics books: Say we have a current density $\vec{j}=\rho\vec{v}$ through a surface $\vec{F}$ of particles $N$ in the volume $V$ with the density $\rho=dN/dV$...
1
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2answers
60 views

Derivative of $x^y=y^x$ defines: $y=y(x)$ [closed]

I need to find the derivative. given that: $$x^y=y^x$$ defines: $$y=y(x)$$ Thank you!
1
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0answers
20 views

How to extend a function to be periodic and smooth?

Assume we have a function f(x) that is twice differentable on [0, L]. Let us define F(x) = f(x) on [0, L], F(x) = -f(-x) on [-L, 0], and F(x + 2L) = F(x) outside of [-L, L]. Thus, F(x) is ...
1
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3answers
36 views

Derivative of $y-2\sin(y)=x$ defines: $y=y(x)$

I need to find the derivative of $y'$ and $y''$ given that: $$y-2\sin(y)=x$$ defines: $$y=y(x)$$ Thank you!
1
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1answer
26 views

Derivatives of $f(x,t)=\varphi (x-at)+\psi (x+at)$

Given that $$f(x,t)=\varphi (x-at)+\psi (x+at)$$ $$u=x-at$$ $$v=x+at$$ We need to prove that: $$\frac{\partial^2 f}{\partial t^2}=a^2\frac{\partial^2 f}{\partial x^2}$$ We know how to calculate the ...
1
vote
2answers
29 views

Given that $f(x)=\frac{1}{x^n}$, show that $x f'(x)+n f'(x)=0$.

This exercise was in my math book and of course had no solution as it's a "show" type of question. I don't see how this could hold except for when $x=-n$. Given that $f(x)=\frac{1}{x^n}$, show that $...
0
votes
1answer
30 views

Is $f'(a)\ge0$ or $f'(a)>0$

If $f$ is smooth s.t. $f<0$ on $(0,a)$ and $f>0$ on $(a,1)$ is then $f'(a)\ge0$ or $f'(a)>0$ ? Is it possible that $f'(a)=0$, maybe you have an example ?
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1answer
23 views

Polynomials bounded on integers

Let $p:\mathbb{R}\rightarrow \mathbb{R}$ be a real valued polynomial, such that for all integers $0\leq i\leq n$ we have $b_{1}\leq p(i)\leq b_{2}$. Let $k=\max_{0\leq x\leq n}|p'(x)|.$ Then for all ...
2
votes
1answer
31 views

Can we say that, there is a neighborhood of $x_0$ such that, $f$ is differentiable in all points of neighborhood?

Let $f:\mathbb{R} \to \mathbb{R} $, and $f$ is differentiable in $x_0$. Can we say that, there is a neighborhood of $x_0$ such that, $f$ is differentiable in all points of this neighborhood? Which ...
0
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3answers
44 views

Find the derivative of the function when given an exponential function

$y=5{x^2}e^{3x}$ Would the rule that I use for this problem be $\frac{d}{dx} e^x=e^x$ We just started learning derivatives of exponential functions and I am a little confused on where to start with ...
0
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1answer
72 views

The total revenue from the sale of x stereos is given by $R(x)=3000(1-\frac{x}{600})^2$. Find the marginal average revenue.

The total revenue from the sale of x stereos is given by $R(x)=3000(1-\frac{x}{600})^2$. Find the marginal average revenue. $R(x)=3000(1-\frac{x}{600})^2$ A. $0.008-(\frac{3000}{x^2})$ B. $5-(\frac{...
0
votes
1answer
16 views

Find all the values of x for the given function where the tangent line is horizontal

f(x)=${\sqrt{x^2+10x+34}}$ The answer choices are A. -5,5 B. 0,-5 C. -5 D. 0,5 Using the chain rule I have found the derivative of this problem as $(2x+10)$${1 \over2(x^2+10x+34)^1/2}$ Given ...
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1answer
80 views

Fréchet differentiability of $\frac{x^3y^2}{x^4+y^4}$ at $(0,0)$?

Suppose a function $f$ is defined as follows: $$f(x,y)=\begin{cases} \frac{x^3y^2}{x^4+y^4}&\text{ when }(x,y)\neq(0,0),\\0 & \text{ when }(x,y)=(0,0).\end{cases}$$ I want to determine ...
0
votes
0answers
24 views

Density of zeroes of the Jacobian of an injective function

I'm looking for a kind of reverse result to the Inverse Function Theorem. Let $f:\mathbb{R}^n\longrightarrow\mathbb{R}^n$ be a differentiable injective function. Is it true that points where the ...
1
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1answer
17 views

Maximum of rectangle in triangle

We have triangle ABC, AB=13cm AC=14cm and BC=15cm. On AC we put a K then AK=x(cm) and we create a rectangle KLMN that is in ABC. Find x for the area of rectangle KLMN is maximum. Sorry for my English ...
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0answers
27 views

Inequality between norm of function and it's derivative

There is a theorem: Let $f$ be a continuously differentiable, $2\pi$-periodic function. Given $\int_{-\pi}^{\pi} f(x) dx = 0$, I need to prove that $$||f|| \le \frac{\pi}{2} \cdot ||f'||.$$ Where ...
0
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0answers
33 views

Proof verification: a set of $10$ times but not $11$ times differentiable functions is not a vector space

I need to find a counterexample showing that the set of $10$ times but not $11$ times partially differentiable functions is not a vector space (under the usual $+$ operator for functions and usual ...
0
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1answer
35 views

Finding the derivative of an indicator function using the limit definition.

Find the derivative of the function: $$\frac {x^2}2 \cdot (I[x \ge 0] - I[x < 0])$$ Using the limit definition: $$\lim_{h \to 0} \frac {f(x+h) - f(x)}{h}$$ Now at a simple glance, I know the ...
1
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3answers
69 views

Finding the second derivative of $f(x) = \frac{4x}{x^2-4}$.

What is the second derivative of $$f(x) = \frac{4x}{x^2-4}?$$ I have tried to use the quotient rule but I can't seem to get the answer.
0
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1answer
15 views

Find the first derivative of given limit

Let $f(x)$ is a polynomial satisfying $f(x).f(y)=f(x) + f(y) +f(xy) -2 $ for all x ,y and $f(2)=1025$ , then the value of lim x tending to 2 $f'(x)$ is I want to know that value at $f(1)=1$ can ...
1
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1answer
20 views

2 ways of solving derivative of composition of functions?

Functions: $f\left(u,v\right)=u^{2}+3v^{2}$ $c\left(t\right)=\begin{pmatrix} e^{t} \\ e^{-t} \end{pmatrix} $ I calculate composition and drivative on 2 ways: 1. substitution and 2. chain rule. ...
2
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1answer
46 views
1
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1answer
40 views

$f \in C[a,b]$ , $f''$ exists in $(a,b)$ ; $\exists t \in (0,1) : f(ta+(1-t)b)=tf(a)+(1-t)f(b)$ ; then $\exists c \in (a,b)$ such that $f''(c)=0$? [closed]

Let $f:[a,b] \to \mathbb R$ be a continuous function , twice differentiable in $(a,b)$ , such that $\exists t \in (0,1)$ such that $f(ta+(1-t)b)=tf(a)+(1-t)f(b)$ ; then is it true that $\exists c \...
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0answers
31 views

Derivative and tall-pipe of truck

Some trucks has vertical tall-pipe with a moving (or fluttering) latch. Am I right that latch's movement is a derivative of accelerator's movement (or amount of exhaust gas leaving pipe)?
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1answer
36 views

Calculate future price with continuous compounding

A forward contract for 4 months is entered into when a stock index is at 1000. If the risk free interest rate is 3% per year (with continuous compounding) and the dividend yield on the index is 2% ...
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0answers
43 views

What is the degree of the differential equation formed from $y= Ax + A^3$

Problem:- What is the degree of the differential equation formed from $y= Ax + A^3$ Solution:- $y= Ax + A^3$ Differentiating both sides we get $y'= A$ Combining both eqations we get $y= y'x + y'^...
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3answers
42 views

Differentiability implies continuity with alternate form

I understand graphically how a differentiability implies continuity. However how can I prove it using this form of the derivative definition: $$ f'(a)=\lim_{h\to 0}\frac{f(a+h) - f(a)}{h} $$
2
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2answers
49 views

How can bring these functions under the integral?

My question is regarding problem 3.5 in Boyd's and Vandenberghe's "Convex Optimization" book. However, I want to ask not about the problem itself but rather about the published sample solution by ...
2
votes
2answers
53 views

Prove that $\frac{dy}{dx} = -\frac1{(1+x)^2}$ for a given equation

$$x\sqrt{1+y} + y\sqrt{1+x} = 0$$ Please tell me where I went wrong. Why I am not getting correct answer ?
2
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3answers
56 views

Finding the second derivative of $x^x$

Find the second derivative $d^2y/dx^2$ when $y=x^x$ ($x > 0$) I found the first derivative, I want to know how to find the second derivative of this function.
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0answers
15 views

Finding the rates of investment of every variable in a multi-variable equation to get the most efficient output.

Math is said to be about teaching problem-solving skills for all problems, but a lot of problems have too many factors to reliably get a result. So how exactly do you get the answer to an equation ...
1
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1answer
44 views

what is $\frac{d}{d(ax)}f(x)$ and $\frac{d}{d(ax)}f(x)$

I have an ODE to which I want to introduce the new variable $\xi=ax$, where $a$ is a constant. How do I calculate the first and second derivatives of some function $f$? $$\frac{d}{d(ax)}f(x),~ \frac{d^...
3
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3answers
95 views

Derivation of $1 = x^2+y^2$ with respect to time [duplicate]

I am studying differential algebraic equations. Given the following equation: $1 = x^2+y^2$ Differentiate this equation with respect to time. The correct solution is: $0=2x \dot x + 2y \dot y $ ...
0
votes
0answers
14 views

Minimize two-variable function

I wish to minimize a function of two variables $m$ and $L$ (both strictly positive). I have calculated the first two partial derivatives: $$\left[ \frac{-n}{2L} + \sum_{i=1}^n \frac{ (x_i - m)^2}{2m^2 ...
0
votes
0answers
4 views

What is the limit formula of a mixed second order partial derivative?

See also this question. Consider following limit formula of the second order partial derivative of a function $f(x,y,z)$: $$\frac{\partial^2f(x,y,z)}{\partial x^2}=\lim_{\Delta x\rightarrow 0}\frac{...
1
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1answer
28 views

How to prove the limit formula of the second order partial derivative?

Consider following limit formula of the second order partial derivative of a function $f(x,y,z)$: $$\frac{\partial^2f(x,y,z)}{\partial x^2}=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x,y,z)-2f(x,...
10
votes
1answer
214 views

Are there parts of Integral Calculus that just *have* to be memorized?

Note : In this question I speak more from a calculation/operational point of view, as opposed to a more theoretical (Analysis) point of view. When studying Differential Calculus, I found that ...