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

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2
votes
2answers
50 views

The slope of a curve when $x = 1$ and $y = \frac{1}{2}$

The surface with equation $z = x^{3} + xy^{2} $ intersects the plane with equation $2x-2y = 1$ in a curve. What is the slope of that curve at $x=1$ and $ y = \frac{1}{2} $ So I put $ x^{3} + xy^{2} = ...
1
vote
1answer
60 views

$n$th derivative of $\frac 1{f(x)}$

Is there a closed-form solution for $\frac {d^n}{dx^n}\frac 1{f(x)}$? I've looked at the first five derivatives in search of some pattern, but I can't identify anything strong enough to give a closed ...
5
votes
6answers
139 views

Find the limit of $\lim\limits_{x\rightarrow0}\frac{x}{\tan x}$.

Find the limit of $\lim\limits_{x\rightarrow0}\frac{x}{\tan x}$ Clearly, since the limit takes the form of $\frac{0}{0}$, one should try L'Hopital's Rule. If we apply L'Hopital's Rule, the problem ...
0
votes
2answers
41 views

solving linear ODEs for mixing problems with function notation

consider this mixing problem for differential equations: a tank holds 100 L of water that initially has no sugar in it. Sugar water with 5 grams / L of sugar enters at rate of 2 L per minute. Water is ...
0
votes
2answers
57 views

Intro to chain rule - find two functions?

I'm learning about the chain rule in class right now. For a homework problem, we have four functions $h(x)$, and we need to find two functions such that $h(x) = f(g(x))$. My question is, how do I find ...
1
vote
1answer
67 views

How to calculate the derivative of a Lie bracket in a coordinate-free setting?

For a given Riemmanian connection defined on a smooth manifold $M$, we denote its covariant derivative by $D_V$ where $V\in \mathcal{x}(M)$, the smooth vector fields on this manifold. Then is it ...
0
votes
1answer
22 views

Converting prime notation of derivatives to Leibniz notation.Resources needed

I have been studying calculus for past few months and through the time I have been using the so called prime notation.I have been studying from Spivaks Calculus for those of you who are familiar with ...
2
votes
0answers
37 views

A continuous funtion with all directional derivatives but NOT differentiable??

I need a function f:R^2->R continuous in R^2, and such that all its directional derivatives exist, but f must not be differentiable in R^2. I know examples of functions continuous but not diff., and ...
1
vote
2answers
30 views

Taylor expansion of the series

function f is given by an equation: $$f(x)=\frac{1}{3+x^3}$$ Find the taylor expansion in a point $x_0=0$ and calculate radiu of the convergence. Could you explain how to find taylor expansion of such ...
2
votes
1answer
81 views

Boundedness of $f'(x)/x$ implies uniform continuity of $f(x)/x$ on $(1,\infty)$

Let $f:(1,\infty) \to \mathbb{R}$ be differentiable, define $g, h:(1,\infty) \to \mathbb{R}$ by $g(x)=f'(x)/x$ and $h(x)=f(x)/x$. Suppose $g$ is bounded. Prove that $h$ is uniformly continuous. I ...
0
votes
1answer
41 views

Max volume of a cone in a sphere

I have given to compute the maximum volume of a cone inscribed in a sphere of radius $R>0$. My question is give that x-radius of the cone the formula to maximize is $\frac{1}{3} \pi ...
0
votes
3answers
37 views

Prove or disprove: If $f$ is continuous and differentiable in $[a,b]$ then $a$ is a local minimum or a maximum point in $[a,b]$.

Prove or disprove: If $f$ is continuous and differentiable in the interval $[a,b]$ then $a$ is a local minimum or a maximum point in $[a,b]$ I'm trying to disprove by giving a counterexample, any ...
1
vote
3answers
39 views

Proof with derivatives (most likely induction)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be given by an equation $$f(x)=(\sin(x^3))^3$$ With use of the fact that function f is odd, show that all derivatives in a form $f^{(2n)}(0)$ for $n=0,1,2, ...
2
votes
1answer
42 views

Derivatives of a function

I came across this problem and was wondering if I could get some guidance with this one? True / False. Every function f that is differentiable on the closed interval [a,b] is itself the derivative of ...
0
votes
1answer
26 views

Derivative Applications Word Problem

When a space shuttle is launched into space, an astronaut's body weight decreases until a state of weightlessness is achieved. The weight $W$ of a 150 pound astronaut at an altitude of $x$ kilometers ...
1
vote
1answer
39 views

Simplification of the derivative

I have this equation $y=x^{5x^3}$ by doing a log transformation we get, $log (y) = 5x^3 log (x)$ upon doing a differentiation w.r.t $(x)$, we get $$\frac{1}{y}\frac{dy}{dx} = 5x^3.\frac{1}{x} + ...
0
votes
1answer
32 views

Composite function derivative/ Chain Rule

If $F(x) = f(g(x))$, $g(2) = 4$, $g'(2) = 3$, $f'(4) = 5$, what is $F'(2)$? Please explain how you got the answer as well.
1
vote
1answer
38 views

Need help to simplify the derivative

Can someone tell me what would be the output of this equation? $$\frac{d}{dx}[\cos^4(x)\cdot\cos (x^4)] = -4x^3\cdot\cos^4(x)\cdot\sin (x^4)+4\cos(x^4)\cdot\cos^3(x)\cdot\sin(x)$$ But am not getting ...
1
vote
0answers
30 views

need help to understand the differential equation

In one of the books, it was mentioned $\frac{d}{dx}(x^3 \tan x)= (x^2\sec^3x+3x^2\tan x)$, but i think it should be $(x^3\sec^2x+3x^2\tan x)$. I feel its a printing mistake. Just wanted to be sure, ...
1
vote
2answers
50 views

find the derivative

$$\cfrac{(x-6)(x^2+4x)}{x^3}$$ and $$\left(\frac{q^6+4}{2q}\right)\left(\frac{q^8+ 6}q\right)$$ Okay so Im reviewing for a test I have tomorrow and these two derivative question come up ive been ...
1
vote
1answer
36 views

Show that the linear function f(x)=Ax is differentiable at a

How to show that the linear function f: $R^{n}$ $\rightarrow$ $R^{n}$, defined by f(x) = Ax, is differentiable at a generic point a, where A is a n $\times$ n matrix, and what is Df(a)? from the ...
1
vote
1answer
28 views

“Global surjective theorem”

So in Multivariate analysis we are doing mapping Theorems and we had one homework problem that I have not been able to solve for over a day now. $ g: \mathbb{R^p \to R^p}\,\text{belongs to class ...
-5
votes
2answers
56 views

derivative /algebra/some help [closed]

i want some help please thank you
0
votes
1answer
24 views

Looking for verification of related rates logic!

I'm working through a really interesting problem about related rates here and I'm pretty sure I've got it figured out. But I'd like to get an expert's opinion on my method here: "Sand is falling ...
0
votes
1answer
35 views

Need help with logarithmic differentiation

I need to use logarithmic differentiation to get f(x)=x$\sqrt{(x+1)(x+2)(x+3)(x+4)}$. I've been working on it for a while and could use some help. Thanks!
2
votes
3answers
58 views

Critical numbers of the function: $x\sqrt{5-x}$

Let f(x) = $$\displaystyle f(x) = x\sqrt{5-x} $$ On the interval: [-6,4] Critical numbers are the the values of x in the domain of f for which f'(x) = 0 or f'(x) is undefined. Derivative of the ...
0
votes
0answers
25 views

Need help in differentiating following problem

I need to find the theta, so the x (distance) is largest. I tried solving it by using differentiation, finding when slope is zero, which should give me the answer. ...
1
vote
1answer
26 views

differentiability of partial derivatives

Prove that if f a function of n variables is continuously differentiable in an open subset U of $R^n$ then the partial derivatives of f are continuously differentiable. I used the definition of f ...
0
votes
2answers
73 views

Prove that if $\lim \limits_{n \to \infty} f(n)=2$ and $\lim \limits_{x \to \infty} f'(x)=0$, then $\lim \limits_{x \to \infty} f(x)=2$.

Let $f: \Bbb{R}\rightarrow\Bbb{R}$ be a differentiable function and let $\lim \limits_{n \to \infty} f(n)=2$ ($n$ runs for Natural numbers), and $\lim \limits_{x \to \infty} f^\prime(x)=0$. Show ...
0
votes
1answer
20 views

finding the values of a such that an implicit function g(y)=x has max,min,saddle points along y=0

I've got $$f(x,y)= a\exp(1+xy) + a^2 \sin(x) +1$$ for which I've shown that there exists an implicit function $x=g(y). ( df/dx \neq 0)$ and $df/dx = a y \exp(1+xy) + a^2 \cos x$ now in the ...
1
vote
0answers
26 views

Continuous second derivative over the support of a Daubechies4 wavelet

I can not entirely follow the proof from section 3.1.1 from the book "A primer on Wavelets" by Walker. After the first part (listed below), I can grasp the rest so if you could help I would greatly ...
2
votes
0answers
23 views

Why doesn't the $L_2$ norm differentiable at $x=0$?

Why doesn't the $L_2$ norm differentiable at $x=0$? Let's define $N(x)$ as the norm function. I know that for every $x\ne 0$: $$\frac{\partial N}{\partial x_i}(x) = \frac{x_i}{\|x\|}$$ What ...
1
vote
1answer
59 views

Derivative of a summation function in order to minimize the function

I'm asked to minimize this function $$f\left(x\right)= \sum_{k=1}^K \left(g\left(w\left(k\right)+\alpha\right)-t\left(k\right)\right)^2$$ with respect only to $\alpha$. Function ...
2
votes
1answer
38 views

Show that $F(x) = f(\|x\|)$ is differentiable on $\mathbb{R}^n$. [duplicate]

Let an even function $f:\mathbb{R}\to\mathbb{R}$ which is even and differentiable. We define $F:\mathbb{R}^n\to\mathbb{R}$ as $F(x) = f(\|x\|)$. Show that $F(x)$ is differentiable on ...
0
votes
1answer
26 views

Show that a function is a contraction in the metric d(x,y) = |lnx - lny|.

We have a function $f: \: (0,\infty) \rightarrow (0,\infty)$, and there is a constant $0<k<1$ s.t. $$x|f'(x)| \leq kf(x).$$ I want to show that $f$ is a contraction. Solving the differantial ...
2
votes
1answer
52 views

Finding all real solutions to the equation $3^x+4^x=5^x$

Find all real solutions to the equation $$3^x+4^x=5^x.$$ My attempt: It is evident that $x=2$ is a solution. However, I think that there are no other solutions. So, I define a function ...
1
vote
2answers
36 views

Showing that if $xf(x)=\log x$ for $x>0$ then $f^{(n)}(1)=(-1)^{n+1}n!\bigg(1+\dfrac12+\dfrac13+\ldots+\dfrac{1}{n}\bigg)$

Let $f(x)$ be a function satisfying $$xf(x)=\log x$$ for $x>0$. Show that $$f^{(n)}(1)=(-1)^{n+1}n!\bigg(1+\dfrac12+\dfrac13+\ldots+\dfrac{1}{n}\bigg),$$ where $f^{(n)}(x)$ denotes the $n$th ...
0
votes
3answers
43 views

How to do the derivative when an exponent has an exponent

I am trying to solve an equation that is in the form of $y(x) = (c + x^2)^{x^2}$. Note $c =$ constant My initial thoughts are I need to look into using ln and e to solve this. However what I am ...
1
vote
3answers
84 views

How to solve an Inverse differentiation problem

If f is a one-to-one function where $f(3)=2$ and $f'(3)=6$, what is the value of $(f^{-1})'(2)$? I am not even sure where to start with this question. I was hoping someone can help $f$ of $3 =2$ and ...
1
vote
1answer
53 views

Related Rates and Angle of elevation

I have been trying to wrap my head around related rates, which are super interesting but very difficult for me personally. Would anyone care to verify if my logic is correct here? "A balloon rises ...
2
votes
1answer
40 views

Rearranging an equation to form the limit definition of derivative

I am following a proof which starts with the following inequalities: $$S_{i}(v) \geq S_{i}(v+dv) + (-dv)P_{i}(v+dv)$$ $$S_{i}(v+dv) \geq S_{i}(v) + (dv)P_{i}(v)$$ From this, we rearrange to form: ...
1
vote
3answers
35 views

Differentiable approximation $f(x) = x$ for $x>0$ and $0$ otherwise.

I would like to find a twice continuously differentiable approximation of $$f(x)= \begin{cases} 0 & x\leq 0 \\ x & x>0. \\ \end{cases}$$ Are there any approximations ...
0
votes
4answers
41 views

Differentiate the following expression: $f(x)=2x^2 + 7x - \ln(x^2+1)$

Differentiate the following expression: $f(x)=2x^2 + 7x - \ln(x^2+1)$ I changed the $\ln(x^2+1)$ to $\frac{1}{x^2+1}$, but I don't know how to solve this.
0
votes
0answers
18 views

To find maximum number of solutions to $f(x) = y$ in (0,1)

Let f : (0,1) to R be a continously differentiable function such that f' has finitely many zeroes in (0,1) and f' changes its sign at exactly two of these points .Then for any y , element of R ,the ...
2
votes
0answers
43 views

Truth of an inequality involving differentials

Is the following inequality true? $$ s\frac{\partial \frac{\partial f(s,t)}{\partial s}}{\partial t}-\frac{\partial f(s,t)}{\partial t}>0 $$ Given that $f(s,t)$ is a monotonically-decreasing ...
3
votes
4answers
98 views

Find derivative of $x^{x^x}$

Trying to find the derivative of: $$ x^{x^x} $$ I have a solution but cannot understand the third transition:
1
vote
2answers
38 views

Differentiable $N$ times, $f'(x_{0})=\cdots=f^{(N-1)}(x_0)=0$, $f^{(N)}(x_0)>0\implies f$ increasing on $x_0$?

Let $I\subseteq \mathbb {R}$ be an open interval and $f:I\rightarrow \mathbb {R}$ is differentiable $N$ times in $x_0\in I$. It's given that: $$f'(x_{0})=\cdots=f^{(N-1)}(x_0)=0, \qquad ...
1
vote
0answers
35 views

Need help to find maximum of this summation equation

I tried to solve secretary problem myself, and I found its bruteforce equation. I am proud that I understood what it is and I can also solve its variations by changing equation. However I don't have ...
0
votes
2answers
39 views

Intuitive meaning of second, third and fourth derivatives at a point.

Can someone explain me the intuitive meaning of second, third and fourth derivatives of a function say, $f(x)$ at a point (say, $a$)? I know it's hard to explain to someone novice like me! But an ...
3
votes
3answers
82 views

How to simplify $y = \frac{\sin\rho + \sin2\rho}{\cos\rho + \cos2\rho}$

How can I simplify this function before I differentiate it? $$y = \frac{\sin\rho + \sin2\rho}{\cos\rho + \cos2\rho}$$ Of course you could immediately start off by using the quotient rule, but that ...