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

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24 views

Derivative of Lattice Laplacian

The lattice Laplacian is defined as, $$ \nabla_L^2x_j \equiv \frac{x_{j+1} - 2x_j + x_{j-1}}{a^2} $$ where the lattice spacing, $a$, is a constant. The derivative, with respect to $x_i$, then gives, ...
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1answer
36 views

How to prove the following function is convex?

I was working on a problem and it reduced to show that $$f(a)=log\Big(\sum_{i=1}^{r}a^ix_i\Big)~~a>1, x_i>0$$ is convex. I have $$f^{\prime \prime}(a)=\frac{\partial^2f(a)}{\partial a^2}=\frac{[\...
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1answer
53 views

Is $f(a+b\epsilon)=f(a)+b\epsilon f'(a)$ true for non-analytic smooth functions of dual argument

Does $f(a+b\epsilon)=f(a)+b\epsilon f'(a)$ remain true for non-analytic smooth functions of dual number argument? Where $ \epsilon^2=0$, $\epsilon \neq 0$ and $a,b \in \mathbb R$. I found proofs ...
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1answer
57 views

Can we say that $f(x)$ exactly one common point with the line segment $y=\alpha x$?

Let $f:(0,1) \to \mathbb{R}$ be a continuous decreasing function, and $\alpha \in {R^ + } - \{ 1\} $? Can we say that $f(x)$ exactly one common point with the line segment $y=\alpha x$ such that $...
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2answers
42 views

Having trouble with partial derivatives

I am having trouble calculating partial derivatives of a simple function. The function is: $$ y(a,b,c)=\frac {0.99821*(a-b)}{c-b} $$ And I need to calculate $ \frac {\partial y}{\partial a} $, $\...
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2answers
29 views

Please explain this differentiation step

I don't get how they went from line 1 to line 2. Which one is treated as the variable and which the constant? I rearrange line 2 to get $0=\frac{3\varepsilon}{M}-h^3$, but I still cannot see how we ...
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1answer
61 views

Existance of a function whose derivative of inverse equals the inverse of the derivative

I've been thinking about the calculation of inverse function through Taylor series expansions. My hypothesis was that if we had $$\ f(x) =\sum_{n=0}^\infty \frac{(x-x_0)}{n!}f^{n}(x_0),$$ then $$\ f^{-...
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1answer
52 views

Directional Derivative defines Descent Direction

Let $f:\mathbb{R}^m \mapsto \mathbb{R}$ be a proper convex function that is not necessarily differentiable and let $x\in\mathbb{R}^n$ be such that $\mathbf{0} \notin \partial f(x)$. I want to prove ...
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1answer
28 views

Is there a derivative for $|x|$ at $0$ specifically “in the direction” of positive $x$?

I know that $|x|$ is not differentiable at $x=0$ because there is potentially an infinite number of tangent lines going through that point. But let's say we were interested in the motion of an object ...
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3answers
267 views

Are derivatives always continuous? [duplicate]

I am assuming first off that the derivative exists everywhere on the real number line (or everywhere in whatever set you choose to work in if for some insane reason you drag complex numbers or ...
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0answers
49 views

Is the $x$-axis a differentiable function? [closed]

Is the $x$-axis a differentiable function?
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3answers
2k views

Derivative/integral relationship appears to disprove the fundamental theorem of calculus!!!

Consider the floor function: $$f(x) = \lfloor x \rfloor$$ The indefinite integral of f is: $$\int_0^x f(x) dx = x\lfloor x \rfloor - \frac {\lfloor x \rfloor^2 + \lfloor x \rfloor} 2$$ This should ...
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0answers
22 views

Show that $f$ defined on the interval $(a,b)$ is not differentiable for every point in $E$ with $m(E)=0$

Let $E$ have measure zero contained in the open interval $(a,b)$. In a previous problem I showed that there is a countable collection of open intervals, $\{(c_k,d_k)\}_k$, contained in $(a,b)$ for ...
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2answers
42 views

$2$ points on a curve have a common tangent

Let $2$ points $(x_1,y_1)$ and $(x_2,y_2)$ on the curve $y=x^4-2x^2-x$ have a common tangent line. Find the value of $|x_1|+|x_2|+|y_1|+|y_2|$. It seems to me that I a missing a link and hence the ...
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1answer
52 views

Calculus 3 - directional deriviative

I recieved the following question: Calculate the directional deriviative at the point (0,0), of the function: $f(x,y) = x^{2}y + xe^{(x-y)}$ and in a direction that is tangent to the curve: $x^{2}...
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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 ...
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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 ...
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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 ...
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3answers
33 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. $...
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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 ...
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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)\...
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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 ...
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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 ...
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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)=\...
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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 $−4^\circ C$...
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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 $\rho=dN/dV$...
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2answers
59 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!
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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 ...
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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!
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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0answers
63 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{...
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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 ...
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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 ...
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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
26 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 ...
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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 ...
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1answer
34 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 ...
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3answers
68 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.
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1answer
14 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 ...
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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. ...
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1answer
46 views