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

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4
votes
4answers
146 views

Proof of an inequality involving $e^x$.

Prove that $e^{x-1} \geq x, $ for every $x$. I'm not allowed to use MVT or integrals, but IVT and derivatives are allowed. I tried to define a function $f(x)=e^{x-1}-x$ and then ...
0
votes
0answers
58 views

Gradient of a complex quadratic form

I have to compute the gradient of the following expression: $$ \nabla_\overline{h} \left( h^H R h - h^H s\right) $$ where the overline means "conjugate of" and $^H$ means conjugate transpose (or ...
1
vote
1answer
84 views

How to write down the second Fréchet-derivative?

I am supposed to express the second Fréchet-Derivative of a function $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ by its partial derivatives. I know how to do this for the first Fréchet-derivative which ...
2
votes
3answers
95 views

What are the partial derivatives of a map $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$

I was wondering what the definition of a partial derivative of such a map is? Is it $\partial_if_j$ or is it anything else? The reason why I have doubts is that I found the definition that a partial ...
15
votes
2answers
465 views

What is the derivative of ${}^xx$

How would one find: $$\frac{\mathrm d}{\mathrm dx}{}^xx?$$ where ${}^ba$ is defined by $${}^ba\stackrel{\mathrm{def}}{=}\underbrace{ a^{a^{\cdot^{\cdot^{\cdot^a}}}}}_{\text{$b$ times}}$$ Work so ...
0
votes
1answer
34 views

Finding equation by given definitions.

In each of the following if there exists a function $f$ that satisfies the given condition, give an example of such a function; otherwise just write DOES NOT EXIST inside the box. No explanation is ...
0
votes
3answers
99 views

Why is $\frac{\partial }{\partial x}\cos(x-y) = -\sin(x-y)$ but $\frac{\partial }{\partial y}\cos(x-y)$ = $\sin(x-y)$?

Why is $\frac{\partial }{\partial x}\cos(x-y) = -\sin(x-y)$ but $\frac{\partial }{\partial y}\cos(x-y)$ = $\sin(x-y)$? (According to wolfarmalpha) Sorry for this question, but as far as I know, ...
2
votes
2answers
245 views

Showing that the function is continuous but not differentiable

Let $$ f(x,y) = \begin{cases} \dfrac{xy}{\sqrt{x^2+y^2}} & \text{if $(x,y)\neq(0,0)$ } \\[2ex] 0 & \text{if $(x,y)=(0,0)$ } \\ \end{cases} $$ Show that this function is continuous but not ...
2
votes
2answers
82 views

Differentiation of exponential function? [closed]

How to solve derivative $\lim_{n\to\infty}e^{{}^n(x)}$ with respective of $x$ ? Here, ${}^n(x)$ is a tetration function $$ {}^n(x)= \begin{cases} x^{[{}^{n-1}(x)]} & \mbox{ if } {\;n>1}\\ x ...
6
votes
1answer
104 views

Simplify $\sum_{n=0}^{N}\binom{N}{n} \frac{a^{N-n}}{n!} \frac{d^n}{dx^n} f(x)$

Simplify the following expression $$S_N = \sum_{n=0}^{N}\binom{N}{n} \frac{a^{N-n}}{n!} \frac{d^n}{dx^n} f(x), $$ where $a$ is a real number and $f(x)$ is an analytic real function. What is $\lim_n ...
1
vote
4answers
128 views

Help with derivative of $y=x^2\sin^5x+x\cos^{-5}x$

Find $y^{\prime}$ of $y=x^2\sin^5x+x\cos^{-5}x$ My try: $\dfrac{d}{dx}(x^2\sin^5x)=x^2(-5\sin^4x)+(2x\sin^5x)$ $\dfrac{d}{dx}(x\cos^{-5}x)=x(-5\cos^{-6}x)+1(\cos^{-5}x)$ This doesn't seem ...
3
votes
1answer
57 views

Weird ordinary differential equation

I am searching for the functions $f,g:[0,\infty) \to [0,\infty)$ which are both increasing, $f'$ is strictly increasing, $g$ is the inverse of $f$ and $$ g(x)^2=f(x)^2\cdot g'(x)$$ My approach was ...
0
votes
1answer
45 views

Given $f(t)$ differentiable. let $u(x,y) = yf(\cos(x-y))$. find $u'_x, u'_y$.

I've never seen a question like this and I'll be happy if you can help me solve it. This is the WHOLE question: (in case it looks wierd) Given function $f(t)$ differentiable in every point in the ...
2
votes
0answers
137 views

Second derivative of a composite function

Say, we have three Banach spaces $X, Y, Z$ and $g:X \to Y, \ \ f:Y \to Z$ are twice (Fréchet) differenciable. The question is: what is $(f \circ g)''$? Since $(f \circ g)'':X \to ...
4
votes
2answers
129 views

$f(x)=\frac{e^x-1}{x}$ and $f(0)=1$. Finding $f''(0)$ rapidly.

Let be $f(x)=\frac{e^x-1}{x}$ and $f(0)=1$. I have to find $f''(0)$. I tried to solve it with finding the second derivative of the function and after that using the L'Hospital rule more times. But ...
2
votes
5answers
191 views

Help with finding tangent to curve at a point

Find an equation for the tangent to the curve at $P\left( \dfrac{\pi}{2},3 \right )$ and the horizontal tangent to the curve at $Q.$ $$y=5+\cot x-2\csc x$$ $y\prime=-\csc ^2 x -2(-\csc x \cot ...
4
votes
3answers
693 views

The difference between $\Delta x$, $\delta x$ and $dx$

$\Delta x$, $\delta x$ and $dx$ are used when talking about slopes and derivatives. But I don't know what the exact difference is between them.
0
votes
5answers
64 views

Help with logarithmic differentiation problems

$\mathbf{(1)}$ Find $y^{\prime}$ of $y=8^{\sqrt x}$ My try: $\ln y=\ln(8)^{\sqrt x}$ $\dfrac{1}{y}y^{\prime}=\sqrt{x}\ln8$ I don't know how to proceed with right side. $\mathbf{(2)}$ Find ...
3
votes
2answers
111 views

Help with $\dfrac{dr}{d\theta}$ for $r=2\sec\theta\csc\theta$

Find $\dfrac{dr}{d\theta}$ for $r=2\sec\theta\csc\theta$. My try: $$r^{\prime}=2[\sec\theta(-\csc\theta\cot\theta)+\csc\theta(\sec\theta\tan\theta)]$$ ...
0
votes
0answers
45 views

Net Differentiation?

How does one cast differential calculus in the context of general point-set topology? Obviously the standard definition of the derivative is expressed in terms of norms on a Banach space, a special ...
0
votes
1answer
36 views

What does it mean that a function continuous in an environment $M_0(x_0,y_0)$?

I saw this term in a theorem, and I don't know what it means. Saw this originally here: If function $z = f(x,y)$ definied around $M_0(x_0, y_0)$ and have partial derivatives $f'_x(x, y), f'_y(x, ...
3
votes
1answer
123 views

prove that $ f(x,y) = \begin{cases} (x^2 + y^2)\sin\frac{1}{x^2 + y^2},& {(x,y) \ne (0,0)} \\ 0, & {(x,y)= (0,0)} \end{cases}$ differentiable.

prove that $ f(x,y) = \begin{cases} (x^2 + y^2)\sin\frac{1}{x^2 + y^2}, & \text{if $(x,y) \ne (0,0)$} \\ 0, & \text{if $(x,y) = (0,0)$} \\ \end{cases}$ differentiable. I tried to use the ...
1
vote
2answers
164 views

Limits involving trigonometric functions $f(x)=\lfloor{x^2 \rfloor} \sin^2(\pi x)$

I was asked to find for what values of x the function is differentiable and write down the derivative. $f(x)=\lfloor{x^2 \rfloor} \sin^2(\pi x)$ for $x \geq 0$. There are two steps: When $x \in ...
3
votes
1answer
154 views

Can't understand how to find a limit $f(x)=\sqrt{|x|}$

I need to find when $f(x)=\sqrt{|x|}$ is differentiable and find the derivative. I found that it's differentiable when $x \neq 0$ and $f'(x)=\frac{1}{2 \sqrt{x}} \ for \ x>0$ and ...
5
votes
3answers
142 views

Derivative of function with 2 variables

I've leart in Calculus 1 that the derivetive of $f(x)$ is: $$\lim_{h\to0} \frac{f(x+h) - f(x)}{h}$$. suppose $f(x,y)$ is a function with 2 variables, does $$f'(x,y) = \lim_{h\to0} \frac{f(x+h, y+h) ...
1
vote
2answers
70 views

Mean Value theorem problem?(inequality)

I'm trying to solve this mean value theorem problem but confused where to start, If $0<a<b$ prove that $(1-\frac{a}{b})<\ln\frac{b}{a}<\frac{b}{a}-1$ Can someone please lend me a ...
1
vote
3answers
157 views

Limits and derivatives - two questions

I was asked to find two limits. Let $f$ be differentiable function at $x=1$ and $f(1)>0$. $$\lim_{n \rightarrow \infty}\left(\frac{f\left(1+\frac{1}{n}\right )}{f(1)} \right)^{\frac{1}{n}}$$ ...
0
votes
3answers
132 views

The speed of the top of a sliding ladder

A $5$m ladder is leaning against a wall. If the bottom of the ladder is pulled along the ground away from the wall at a constant rate of $0.4$m/s, how fast will the top of the ladder be moving down ...
2
votes
3answers
104 views

On the nature of a first derivative

This is a very, very basic question. Never been very involved in math but I've been learning calculus in my free time, so here goes. I have observed some various things that happen with derivatives, ...
1
vote
1answer
46 views

A question about covariant derivative (find $D_{v_p}W$)

the quest. on my book is $(y_1,y_2,y_3)\quad R^3\quad coordinate \quad system$ . if $W=y_1y_2^2\frac{\partial}{\partial y_1}+(y_3-y_2^2)\frac{\partial}{\partial y_2}+3y_1\frac{\partial}{\partial ...
-1
votes
3answers
141 views

Details of $z=u \cos(v) \sin(u), u=e^{xy^2}, v=x^2+y \quad \frac {\partial z} {\partial x}=?,\frac {\partial z} {\partial y}=?$ [closed]

Given $$z=u \cos(v) \sin(u), \quad u=e^{xy^2}, \quad v=x^2+y$$ How to show details of finding $\dfrac {\partial z} {\partial x}$ and $\dfrac {\partial z} {\partial y}$?
12
votes
2answers
299 views

Finding the derivative of $x\uparrow\uparrow n$

I am trying to find a general derivative for the function: $f(x)=x^{x^{x^{...^{x}}}}$however to do that I must find $f^{\prime }$ and $f^{\prime \prime}$...etc. I am now trying to write down a general ...
1
vote
1answer
72 views

Derivative polynomial [duplicate]

Let $f :\mathbb R\to\mathbb R$ be an infinitely differentiable function. Assume that for every $x \in \mathbb R$, there exists an $n_x\in \mathbb N$, such that $\,f^{(n_x)}(x)=0$. Prove that $f$ is ...
0
votes
4answers
83 views

What is the derivative of the given function

This is the question: What is the derivative of this function? $$\frac{2^{x}}{e^{x}}$$ I have seen two answers for this question and I would like to know which one is correct and which one is ...
1
vote
1answer
62 views

Finding the derivative and stationary point

I had no idea how to attempt this question, but i came across a recap of a lecture and i kinda followed it through. I don't exactly know what i've done as it was really easy to substitute their ...
3
votes
3answers
166 views

General Leibniz rule for triple products

I have a question regarding the General Leibniz rule which is the rule for the $n^{th}$ derivative of a product and reads: $$ (f g)^{(n)}=\sum_{k=0}^{n} {n \choose k} \,f^{(k)} g^{(n-k)}. $$ ...
0
votes
3answers
71 views

what is the derivative of this function

I need help in this following question. I have tried many attempts but really confused on how to solve it. Therefore, i would appreciate any help from you guys. Thanks allot $$f(x) = \dfrac{\ln ...
7
votes
3answers
274 views

How prove this $F'(x_{0})=f(x_{0})$ if $F(x)=\int_{a}^{x}f(t)dt,x\in[a,b]$

Question: let $f$ be Riemann integrable on $[a,b]$,Assmue that $x_{0}\in [a,b]$,and let $f(x)$ is continuous on point $x=x_{0}$,and define $$F(x)=\int_{a}^{x}f(t)dt,x\in[a,b]$$ show that ...
20
votes
1answer
382 views

$n^{th}$ derivative of a tetration function

I stumbled upon this very peculiar function last summer, namely: $f(x)=x^{x^{x^{...^{x}}}}$, where there is a number $n$ of $x$'s in the exponent, I tried to find the derivative for the function and I ...
-1
votes
2answers
88 views

Differential Calculus, Slope at a Given Point [closed]

If $$3x^2 + 2xy + y^2 = 2$$ then what is the value of $dy/dx$ when $x = 1$?
1
vote
1answer
103 views

Inverse function of $y=2x+\sin x$

I was doing a long exercise when come to this point: calculate the inverse function of $y=2x+\sin x (x \in\mathbb R) $ and its derivative. I know that the derivative of an inverse function is ...
3
votes
1answer
74 views

Derivatives and average velocities

It is easily proven that given an everywhere differentiable function $f$ on $\Bbb R$, if $f$ is constant, linear, or a quadratic function, then $$\frac{f(x)-f(y)}{x-y}=\frac{f'(x)+f'(y)}{2}$$ for all ...
0
votes
2answers
34 views

Constructing an antiderivative of a function if the contour integral depends on initial and final point

I am working on the following problem: Let $D \subset \mathbb C$ be a domain, $f: D \to \mathbb C$ a continuous function and $\gamma : [\alpha, \beta] \to D$ a contour. Assume that $\int_\gamma f$ ...
0
votes
1answer
42 views

A problem about a calculus equivalence of inflection point

The problem: Given $f: D \rightarrow \mathbb{R}$ a differentiable function on the interval $(a,b)$, and $g: D \rightarrow \mathbb{R}$ satisfying: $$g(x) = \begin{cases}\dfrac{f(x)-f(x_0)}{x-x_0} ...
2
votes
1answer
128 views

Differentiability in metric spaces

I have a question in mind: Why can't we define differentiability in arbitrary metric spaces? Or can we define it really? Please discuss. I only have studied the notion of differentiability in ...
0
votes
1answer
69 views

Why does the derivate of (x+1)^2 not equal 2?

$$f(x) = (x+1)^2$$ $$f'(x) = 2(x+1)$$ Shouldnt it equal 2 because, the rules of derivation says that the derivation of any number equals 0 the derivation of x^2=2x*1=2 $$(x+1)^2 =x^2+1$$ ...
1
vote
1answer
40 views

Derive $\cos x(\ln x-x)^2$

I have a doubt in deriving this function $\cos x(\ln x-x)^2$, deriving following the rules I obtain $$-\sin x(\ln x-x)^2+\cos x\left[2(\ln x-x)\left(\frac{1}{x}-1\right)\right]$$ Is it correct? I ...
12
votes
3answers
290 views

Simplified form for $\frac{\operatorname d^n}{\operatorname dx^n}\left(\frac{x}{e^x-1}\right)$?

I have found the following formula: $$\frac{\operatorname d^n}{\operatorname ...
1
vote
4answers
42 views

Finding the derivative using quotient rule…

$$\frac{\text{d}}{\text{dt}}\dfrac{2(t+2)^2}{(t-2)^2}$$ I applied the quotient rule: $$\dfrac{[2(t+2)^2]'(t-2)^2-2(t+2)^2[(t-2)^2]'}{(t-2)^4}$$ $$\dfrac{4(t+2)(t-2)^2-2(t+2)^22(t-2)}{(t-2)^4}$$ ...
1
vote
3answers
75 views

How to evaluate this limit with l'hopital's rule

is it possible to use L'hopital for this or is there another method I'm missing? I have no idea how to even start this. $$\lim_{x\to \infty} \frac{(9x+1)^\frac12}{x+1} $$