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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3answers
71 views

How can I differentiate $(ye^x)^{\frac{1}{x}}=y^2$?

I have following relation to differentiate: $$(ye^x)^{\frac{1}{x}}=y^2.$$ However, I got a bit confused: I first simplified: $y^{\frac{1}{x}}e^1=y^2$ and then differentiated, but that doesn't seems ...
0
votes
3answers
50 views

Differentiation proof obyinduction

If we have two functions, $a(x)$ and $b(x)$ and those function are differentiable inifitely many times. What is a closed form to $$\frac{d^n}{dx^n} (ab)$$ How can I use induction here? I don't ...
1
vote
3answers
115 views

Compute $f^{(22)}(0)$ where $f(x)= \sin(x)/x$ if $x\neq0$ and $1$ if $x=0.$

Let $$f(x)= \begin{cases} \frac{\sin(x)}x &\text{ if x}\neq0\\ 1 &\text{ if x}=0. \end{cases} $$ What is $f^{(22)}(0)$? First I found that $$ f'(0)=\lim_{h \to 0}\frac{\sin(h)/h-1}{h} ...
3
votes
3answers
71 views

How to show $\,f(x)=3e^{2x} -10x -7x^2\,$ has a minimum on $\,[0, 1]$

I have been told that $$f(x)=3e^{2x} -10x -7x^2$$ and I need to show that it has a local minimum on the interval $[0,1]$. How would you show this?
0
votes
1answer
50 views

How to prove $(2^{-1/y}(1-x)+x)^{-y}$ is increasing in $y$, when $x,y \in (0,1)$.

As the title suggests, how to prove $(2^{-1/y}(1-x)+x)^{-y}$ is increasing in $y$ when $x,y \in (0,1)$?
1
vote
1answer
40 views

$\int_\Omega |\nabla u^+|^2 \, dx$ is not differentiable with respect to $u$ in $W_0^{1,2}(\Omega)$

Let $u \in W_0^{1,2}(\Omega)$, where $\Omega$ is some domain in $\mathbb{R}^N$, $N \geq 1$. Denote $u^+ := \max\{u, 0\}$. (It is know that $u^+$ also belongs to $W_0^{1,2}(\Omega)$ (see, e.g., ...
1
vote
2answers
60 views

Where does $r = 1 + 2\cos(\theta)$ have tangents?

Where does: $$r = 1 + 2\cos(\theta)$$ Have horizontal and vertical tangent lines? $x = r\cos(\theta) = \cos(\theta) + 2\cos^2(\theta)$ $y = r\sin(\theta) = \sin(\theta) + ...
0
votes
2answers
19 views

Express $\frac{\partial^{2}z}{\partial r^{2}}$ in terms of r, $\theta$, and the partial derivative of f.

Let $z=f(x,y)$ and let r and $\theta$ be polar coordinates in the x-y plane. Recall that $x=r \cos \theta$ and $y=r \sin \theta$. Express $\frac{\partial^{2}z}{\partial r^{2}}$ in terms of r, ...
0
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0answers
53 views

Study $f_{\lambda}(x) = \lambda e^x + x^2 + 2x +2$ for any $\lambda \in \mathbb{R}$

This time I have the following questions: Consider $$f_\lambda: x \longmapsto \lambda\exp(x)+x^2 +2x +2$$ for any real $\lambda.$ 1) Compute $f'_\lambda$ (the derivative of $f_\lambda$). Show ...
0
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0answers
34 views

Determine if the graph $f(x) = \ln(x)$ has any critical numbers

Determine if the graph $f(x) = \ln(x)$ has any critical numbers. The derivative would be $f'(x) = \frac{1}{x}$
-1
votes
2answers
92 views

A question from Analysis: Differentiation

I have not been able to solve this specific question pertaining to differentiation for the course real analysis. How would you go about on this one? Show that if $f^{(n)}(x_0)$ and $g^{(n)}(x_0)$ ...
0
votes
1answer
12 views

If a function $f(x,t)$ is globally Lipschitz, does that implies that it is continuously differentiable in $x$?

I know that it is necessary that $df/dx$ continuously exists and it has to be uniformly bounded for $f(x,t)$ to be globally Lipschitz. But, if a function $f(x,t)$ is globally Lipschitz, does that ...
0
votes
3answers
52 views

Find the partial derivatives of the following: $f(x,y,z)=x^{\sin(y^{x})}+\int_{0}^{x} t^tdt$.

$f(x,y,z)=x^{\sin(y^{x})}+\int_{0}^{x} t^tdt$. Im not sure how to treat this integral in relation to the different variables.. and the first part also is unclear. $x^{\sin(y^{x})}$ :D
1
vote
1answer
35 views

Derivative of convolution is the convolution with a derivative

I am trying to solve this exercise: Let $\alpha$ be a multi-index. Show that $\partial^{\alpha}(u * v)=(\partial^{\alpha}u)*v$, where $u\in C_0^{k}(\mathbb{R}^n)$ and $v\in L^1_{loc}(\mathbb{R}^n)$. ...
2
votes
2answers
33 views

Mean value theorem with trigonometric functions

Let $f(x) = 2\arctan(x) + \arcsin\left(\frac{2x}{1+x^2}\right)$ Show that $f(x)$ is defined for every $ x\ge 1$ Calculate $f'(x)$ within this range Conclude that $f(x) = \pi$ for every $ x\ge 1$ ...
2
votes
1answer
45 views

if f(x) if differentiable and continuous, prove $\frac{af(a)-bf(b)}{a-b} = f(c) + cf'(c) $

Let $f(x)$ be differentiable at $(a,b)$ and continuous at $[a,b]$ prove that there exist $ c \in (a,b)$ such that: $$ \frac{af(a)-bf(b)}{a-b} = f(c) + cf'(c) $$ I started with: let $c \in (a,b) $ ...
0
votes
1answer
53 views

Find the equation of the line tangent to the following curve at $x=\pi/2$

$$ f(x) = \int_0^{2\pi} \frac{\sin t}{t} dt $$ So far I can find the slope (which I think is zero), but I cannot find the term $b$ of $g(x)=mx+b$.
1
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3answers
51 views

if $f'' >0$ prove that : $f(x+2)-f(x) \le f(x+5)-f(x+3)$

Let $f: R \to R$, differentiable twice such that $f'' > 0$ Prove that for every $ x>0$ exists: $f(x+2)-f(x) \le f(x+5)-f(x+3)$ Any hints/suggestions? I got this problem at class and I ...
0
votes
1answer
25 views

Derivative of integral with infinity as upper bound

What is the solution to the derivative of following integral? I know how to take derivatives of integrals but I never came across one with infinity in one of his bounds. $F(t) = \int^{\infty}_t ...
1
vote
1answer
81 views

Prove: if $f'(x)$ equals zero once, $f(x)$ equals zero twice

Given $f(x)$ that is differentiable for every $ x \in R$, if $f'(x) = 0$ only once, $f(x)=0$ only twice. I'm having a bit trouble with this proof, I first said that Let $c \in R$ such that ...
0
votes
0answers
16 views

Derivative expected value

Let $f(x) := E_\omega [(\omega x - 1)^-]$, $x \in \mathbb{R}_+$, where $\omega$ is a r.v. from the uniform distribution, i.e., $\omega \sim U(0,1)$. It is obvious to see that the function $f$ is ...
2
votes
2answers
41 views

Integrate via substitution and derivation rule

i have to solve this integral $$\int_{-r}^{+r}\int_{-\sqrt{r^2-x^2}}^{+\sqrt{r^2-x^2}} \sqrt{1-\frac{x^2+y^2}{x^2+y^2-r^2}} \operatorname d y \operatorname d x$$ with substitution and then the ...
3
votes
2answers
75 views

Examples of bounded continuous functions which are not differentiable

Most often examples given for bounded continuous functions which are not differentiable anywhere are fractals.If we include probabilistic fractals exact self-similarity is not required. Are their ...
-1
votes
2answers
106 views

Values of $f(\pi/4)$ and $f'(\pi/4)$ if $\int_0^xf(t)dt=\frac{-1}{2}+x^2+x\sin(2x)+\frac{1}{2}\cos (2x)$ [on hold]

Let $f:\Bbb R\to \Bbb R$ so that for all $x>0$ $$\int_0^xf(t)dt=\frac{-1}{2}+x^2+x\sin(2x)+\frac{1}{2}\cos (2x)$$ Calcule $f(\pi/4)$ and $f'(\pi/4)$.
0
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2answers
25 views

Differentiate folowing expression, how much simplifying?

I have following task: Differentiate $$\sec(\sqrt{x})\cdot\tan\left(\frac{3}{3x+1}\right)$$ I got following result: However, differentiation is new to me and I'm not sure what I have to do ...
2
votes
1answer
26 views

Evaluate a Heaviside step function

So I have this question that I don't know how to do. The Heaviside step function H(x) (also called unit step function) is a discontinuous function whose value is 0 for negative x and 1 ...
1
vote
0answers
52 views

Computing the fractional derivative of a fractional integral

I know that $D^{\alpha}I^{\alpha}f(x)=f(x)$ and $D^{\alpha}I^{\beta}f(x)=D^{\alpha-\beta}f(x)$ but How can prove this? ...
1
vote
2answers
37 views

Using the chain rule backwards

I'm asking about a use of the chain rule that I've seen in a couple of derivations but that I don't understand, I hoping for it to be clarified. Let's say we start with the gravitational ...
6
votes
2answers
59 views

Finding the derivative $f(x)=\sqrt{x^2 -9}$,

I need to find the slope at a=5, using the definition for the function $f(x)=\sqrt{x^2 -9}$, $$f'(x) = \lim_{\Delta x \to 0} {f(x+\Delta x)\over \Delta x}$$ The answer book says the slope is ...
1
vote
2answers
56 views

Finding derivative $f(x)={2\over x^3}$

I have to find the derivative and the slope at $a=6$ The function is $f(x)={2\over x^3}$ I have to find the answer using the formula, $$f'(x)= \lim_{\Delta x \to 0} {f(x+ \Delta x) - f(x) \over ...
1
vote
1answer
50 views

Compute $\int_cd\omega$ and $\int_{\partial c}\omega$

Question: Let $c:I^2\rightarrow\mathbb{R}^3$ be the singular $2$-cube given by $$c(s,t)=\left(\frac{1}{2}s^2,st,\frac{1}{2}t^2\right)$$Let $x=(x,y,z)$ denote the cartesian coordinates on ...
1
vote
1answer
29 views

Find the point where the slope changes drastically

I have a distribution for which I have to find the point where the slope changes drastically. In visual terms, I have to find this point: I though I could use derivatives, but for the following ...
4
votes
2answers
51 views

Question about a differentiable function at point $a$.

Let $f$ be differentiable at point $a$. Prove than if $\lim \limits_{n \to \infty}x_n =\ a^{+}$ and $\lim \limits_{n \to \infty}y_n = a^{-}$ then $$\lim \limits_{n \to \infty} \frac{ f(x_n) - ...
-1
votes
3answers
38 views

Some confusing differentiation questions! [closed]

Given $x =\ a \,\text{sin}\ t - b \sin($$\frac{at}{b})$ and $y =\ a \,\text{cos}\ t - b \cos($$\frac{at}{b})$how to find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ in terms of $t$? How to show that ...
1
vote
3answers
33 views

Investigating monotone and bounded nature of a function.

If $$f(x)=x^3+bx^2+cx+d$$ and $0<b^2<c$, then $f(x)$ in $(-\infty,\infty)$ is increasing is decreasing is bounded has real maximum I solved till $f'(x)=3x^2+2bx+c$ and $f ''(x)=6x+2b$, now ...
1
vote
3answers
75 views

Determining slope that cuts off least area

So here is the question: Determine the slope of the line that passes through the point $(1,2)$ and that cuts off the least area from the first quadrant. I've thought about this question, and I ...
1
vote
1answer
21 views

monotonicity by examining the sign of derivative

given : $x>0 ,y >0 ,b>a>0$ prove the following by using derivative of a appropriate function: $${(x^b+y^b)}^{(1/b)} < {(x^a+y^a)}^{(1/a)}$$ I tried using $f(x)=(m^x+n^x)^{(1/x)}$ and ...
0
votes
2answers
25 views

Extending the derivative to a boundary point

Let $f: [0, 1] → ℝ$ be a continuous function and continuously differentiable on the interior $(0, 1)$. Assume furthermore that $\lim_{x→0}f'(x) ≕ D$ exists. Then $f$ is right-differentiable at $0$ and ...
2
votes
2answers
77 views

Derivative of a vector

Let $p, v :$ real, positive $1\times n$ vectors, $c^T:$ real, non - negative $n\times 1$ vector, $I:$ the identity matrix. Assume that the following relationship holds true: $$p(v) = v\cdot ( I - ...
6
votes
2answers
642 views

When is differentiating an equation valid?

I wonder that Is it true to differentiate an equation side by side. Under which conditions can I differentiate both sides. For example, for the simple equality $x=3$, Is ıt valid to differentiate both ...
0
votes
1answer
26 views

Taylor expansion of $f(z)=\frac{z-1}{z^2-3z+3}$

We are given the function $f: \mathbb C \to \mathbb C$ defined by $f(z)=\frac{z-1}{z^2-3z+3}$ Is it possible to define $f$ as its taylor expansion near the point $z=i\sqrt 3$? If so, what is the ...
0
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0answers
31 views

mathematical anaysis problem [duplicate]

Let $f$ have a finite derivative on $(a,∞)$. If $f(x) \rightarrow 1$ and $f'(x) \rightarrow c$ as $x \rightarrow ∞,$ show that $c=0$. It seems easy question, but unfortunately, I could ...
0
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0answers
31 views

Second derivative of function in matrix form

Given an equation \begin{equation*} log(L_{c}(n|Z^{*}n)) =log\left(\frac{\displaystyle\prod_{k=1}^{K}\frac{(m_{k}^{*})^{n_{k}}\exp(-m_{k}^{*})}{n_{k}!}} ...
2
votes
3answers
49 views

Finding Derivatives $f(x)={1\over x+1}$

I'm using the Limit Definition to find the derivative, $$f'(x)=\lim_{\Delta x \to 0} {f(x+\Delta x) - f(x) \over \Delta x}$$ $$$$ Now, I want to find the derivative for the function, $$f(x)={1 \over ...
0
votes
0answers
9 views

Show that the function $B(A,y)=Ay : M^{3x3}xR^3 \to R^3$, where $M^{3x3}$ is the space of square matrices 3x3, is a bounded bi-linear function.

And also find the first derivative of the function. $\|B(A,y)\| \over {\|A\|\|y\|}$$\leq 1$ therefore bounded? And first derivative$ B(x+H1,y+h2)-B(x,y)= A*h2 + H1*y+H1h2$ now i thnik i can take ...
0
votes
2answers
36 views

Prove that f(x)=C1sinx +C2cosx for constant C1 and C2…

It's given that f is differentiable twice and that $f''+f=0$ I have to show that $f(x)=C_1\sin(x) +C_2\cos(x)$ for constant $C_1$ and $C_2$. There is also a hint: using the given data, prove that ...
1
vote
2answers
61 views

prove $f$ is a constant [duplicate]

Lets's say we have a differentiable function $f:[a,b]\to \mathbb{R}$ with $f^\prime\equiv0$ How do I show that $f\equiv C$ by using the mean value theorem?
-3
votes
2answers
59 views

Find the rate of change $dy/dx$ for $X =X_0$ [closed]

Find the rate of change $dy/dx$ for $X =X_0$ in the following cases... [I don't know if my thoughts are correct. Can someone please help me with this problem.] 1) $y = 3$; $\;X_0 = 2$ 2) $y = ...
0
votes
4answers
61 views

$k'=k$ only for $e^x$ [duplicate]

How can one prove without using anything but differentiation, that $e^x$ is the only function with $f'=f$? Clearly I can prove that $(e^x)'=e^x$, and $0'=0$, but how can one show that no other ...
0
votes
0answers
30 views

How to solve this “Coupled Nonlinear Differential Equations”

First, I'm sorry to bring the formula as picture. I'm not that experienced with formula tags in written. BTW, I need your help with solving this "coupled nonlinear partial differential equations". ...