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

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2
votes
1answer
50 views

Partial derivative with respect to intermediate variable

Suppose we have $f(x,y,z)$, where $x=g(r,\theta,\phi)$, $y=h(r,\theta,\phi)$, $z=t(r,\theta,\phi)$. How do we find partial derivative with respect to $x$ and express it as a "function" of $r$, ...
3
votes
2answers
94 views

Find an equivalent of this function,

a) $f$ continuous on $[0,1]$ such that $f(x)>0$. Find an equivalent of $$h(\epsilon) = \int_0^1 \frac {f(x)}{x^2 + \epsilon^2}dx$$ when $\epsilon$ goes to zero and when $\epsilon$ goes to ...
2
votes
1answer
28 views

Proof of Liouville's formula , details and confusions. [Matrices, determinants..]

So I've got the homogeneous linear equation: $$x^{(n)}+a_1(t)x^{(n-1)}+...+a_{n-1}(t)x'+a_n(t)x=0.$$ where $a_1(t)...a_n(t)$ are real continuous on intervals. This is what my textbook states: If ...
2
votes
1answer
29 views

Calculating the Lie algebra representation of the regular representation on subspace of functions on $\mathbb R$.

Let $G = \mathbb R$ and let $\pi$ be the regular representation of $G$ on $L^2(\mathbb R)$, that is, $\pi(g)(f)(x) = f(x-g)$ for $g \in G$. Let $V = \{f \in \mathcal C_c^\infty | supp f \subseteq ...
0
votes
1answer
63 views

Is there a function that doesn't have a limit at infinity but its derivative does?

Is there a function that is differentiable in $\mathbb{R}$, its derivative converges to $0$ as $x\to \infty$ but the function does not converge as $x \to \infty$, neither to a finite limit nor to an ...
0
votes
1answer
12 views

Derivatives expression understanding problem

From book about hydraulics I saw while reading: $$ \frac{du_x}{dt} = \frac{\partial u_x}{\partial t} + \frac{\partial u_x}{\partial x} u_x +\frac{\partial u_x}{\partial y} u_y +\frac{\partial ...
1
vote
2answers
49 views

Differentiation of complex valued functions. $i^x$

The question is how to find the derivative of $i^x$ or even if it exists?. WolframAlpha does give an answer. Before applying the definition, we see that how can we mix the complex and the real plane. ...
0
votes
1answer
42 views

What's meaning of a derivative to its original funciton?

I have trouble of finding the relationship of a function's derivative with its original function. Suppose there is a function: $$s = f(x)=16x^2 + 2$$ So its derivative is $$s'=f'(x)=32x$$ Here's ...
2
votes
1answer
74 views

Is a function a derivative?

I'm reading introductory calculus and I find that 'function' tends to be defined by what it does rather than what it is. If $y = f(x)$, then surely the value of $y$ is dependent on that of $x$, i.e. ...
2
votes
2answers
61 views

Integrating a First Order Differential Equation (The West Equation)

I am currently doing a project about Growth and have found this really interesting Math Model by Dr. Geoffrey West et al in 2001 while researching. The paper can be found at this link. I was ...
0
votes
1answer
31 views

Is there a concept called the cross derivative between two functions?

Let $f$ and $g$ be two real functions. Is there already a concept for the quantity $\lim_{h \to 0} \frac{f(x+h)-g(x)}{h}$? Note that when $g=f$, the quantity, if exists, is the derivative of $f$ at ...
1
vote
2answers
62 views

Find $y'$ if $y=e^{-4x} \sin\ 5x$

$$y=e^{-4x} \sin\ 5x$$ My answer is: $y' = e^{-4x}(\cos\ 5x)(5)+(\sin\ 5x)e^{-4x}$ = $$e^{-4x}((\cos\ 5x)(5)+(\sin\ 5x))$$ The books answer is different am I right?
1
vote
8answers
149 views

Differentiate the Function: $y=e^{\tan x}$

$y=e^{\tan x}$ The book says to use the Chain Rule. Let $u = \tan x$. Thus, $y = e^u$ $du = \sec^2x\ dx$ $\frac{du}{\sec^2x}= dx$ I am confused at this point. The book explains the method of ...
1
vote
2answers
31 views

Having 2 functions of the same variable, how can I find the derivative of the first function in relation to the other?

Let's be specific and use a simpler example than what I actually need to solve. $$ \begin{split} x(t) &= t + A\sin(wt) \\ y(t) &= B \cos(wt) \end{split} $$ How would I obtain the derivative of ...
3
votes
2answers
320 views

Can the following trick be expanded upon?

Main Question What is the expansion of $d^{1+\epsilon}?$ Background I noticed the following trick (sometimes more laborious) to directly differentiate $ f(x) $ twice without differentiating it even ...
3
votes
2answers
87 views

Show $ \lim_{n \to \infty} \frac{\ f(y_n)-f(x_n)}{ y_n-x_n} = f'(x_0)$ using Taylor

Let $f:[a, b]\to R$ differentiable at $a<x_0<b$. Using taylor series show that if $x_n \to x_0^-$ and $y_n \to x_0^+$ then $$ \lim_{n \to \infty} \frac{\ f(y_n)-f(x_n)}{ y_n-x_n} = f'(x_0)$$ ...
1
vote
1answer
39 views

Derivative of integral and variable substitution

My question is about the validity of this identity and if there is some error in my argument: $$\int_0^{\infty}\frac{d}{dt}f(t-x)dx = -\int_0^{\infty}\frac{d}{dx}f(t-x)dx$$ The argument goes as ...
7
votes
1answer
60 views

Semantics of Writing Differential Equations

Let $f : \mathbb R \to \mathbb R$, and consider the differential equation $$ f'(t) = f(t) $$ it is easily seen that it has the solutions $f(t) = a\cdot \exp(t)$ for $a \in \mathbb R$. Now another way ...
3
votes
2answers
38 views

If $f$ is differentiable in $[a,b]$ and $f'(x) \ne 0$ for each $x \in [a,b]$ so $f$ is monotone in $[a,b]$?

If $f$ is differentiable in $[a,b]$ and $f'(x) \ne 0$ for each $x \in [a,b]$ so $f$ is monotone in $[a,b]$. Is this correct? I don't think so. Because the differntial doesn't have to be continuous so ...
1
vote
3answers
36 views

Derivation of the following function

We know that if $\sin x$ is positive, then $\sin x\le 1$. So $\ln(\sin x)$ is negative. Hence the domain of $f(x)=\sqrt{\ln(\sin x)}$ is $\{2k\pi+\pi/2;k\in \Bbb Z\}$. Therefore ...
2
votes
1answer
49 views

Derivative to Zero, What does it intuitively mean?

I'm currently learning machine learning, and I came across this equation called Least Squares Regression. X and w are both matrices. The multiplication of both matrices becomes y hat, which is ...
0
votes
2answers
30 views

Differentiating product and squares of logarithms

I need help differentiating. I am really confused how to solve with the $\ln x$ in the equation. Which of the logarithm rules do I need to use for this equation? $$y= 12x \ln x + 12x - 6x (\ln x)^2 + ...
0
votes
1answer
19 views

Conditions for functions to be independent of one of their variables

I'm working independently through Spivak's Calculus on Manifolds and I've come across a stumbling block with respect to two of his questions. The first question is 2.22. If $f:\mathbb ...
1
vote
1answer
37 views

chain-rule application

Consider $f:\mathbb{R}^3\to\mathbb{R},(x,y,z)\mapsto x+y+z$ and a differentiable function $g:\mathbb{R}^2\to \mathbb{R}$. What is correctly if I want to apply the chain rule, ...
0
votes
2answers
32 views

Applying the chain rule correctly to $f(x,g(x))$

Consider a function $f:\mathbb{R}^2\to\mathbb{R},\; (x,y)\mapsto f(x,y)$, $g$ and $f$ continuously differentiable, $g:\mathbb{R}\to\mathbb{R}$. How to apply the chain rule on $f(x,g(x))$ correctly? ...
3
votes
3answers
57 views

Let $f:\mathbb{R} \to \mathbb{R}$ be $C^3$. Show the equivalence: $f^{(k)}(0) = 0$ for $k=0,1,2 \iff \lim_{x\to 0} \frac{f(x)}{x^3}$ exists

Let $f:\mathbb{R} \to \mathbb{R}$ be $C^3$. Show the equivalence: $$f^{(k)}(0) = 0 \quad k=0,1,2 \iff \lim_{x\to 0} \frac{f(x)}{x^3} \text{, exists.}$$ Trying: Since $f \in C^3$, implies $f, f', ...
3
votes
0answers
68 views

Show that such an $f$ cannot exist

Suppose $f:\mathbb R^n\to\mathbb R$ is a scalar field, such that for a given vector $a\in\mathbb R^n$ and any $y\in\mathbb R^n-\{0\}$ we have, $f'(a;y)>0$. Show that such a function $f$ cannot ...
0
votes
1answer
46 views

Differentiability issue with this function

$f:D\to{R}$ $$f(x)=\frac{1}{x-2}e^{\left|x\right|}$$ Find the domain $D$ of the function and study whether the function is differentiable. Find the left and right derivatives in the points where the ...
1
vote
3answers
61 views

Indeterminate form $0^0$ using L'hospitals rule when calculating $\lim_{x\to0^+} x^{\sin(x)}$

Given the question $$\lim_{x\to0^+} x^{\sin(x)}$$ I have deducted so far that this has the indeterminate form $0^0$ so I have taken the natural logarithm of both sides to give me: $$\lim_{x\to0^+} ...
5
votes
2answers
61 views

Relationship between $\sin(a+b)$ and derivative product rule?

I noticed this interesting correlation between the sine angle addition formula and the derivative product rule. The sine addition formula is $$\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$$ The ...
9
votes
1answer
114 views

What is the Exterior Derivative Trying to Do?

$\newcommand{\R}{\mathbf R}$ Consider a smooth function $f:\R^n\to\R$ and let $Df:\R^n\to \R^{n*}$ be the map which takes a point $\mathbf a\in R^n$ to the linear map $Df_{\mathbf a}:\R^n\to \R$. ...
0
votes
1answer
33 views

Edited: Implicit Differentiation of Life-History Function

I am trying to implicitly differentiate the following function: $$ \lambda = \exp \left[ \left( \alpha + \frac{s}{\lambda-s} \right)^{-1} \right] $$ Can someone help me with this?
1
vote
2answers
41 views

Solving for y' in a fraction

Given the equation $x+xy^2 = \tan^{-1}(x^2y)$ find $y'$. I have tried doing this but solving for $y'$ I need some help and would like your advice. Work so far... ...
3
votes
1answer
69 views

How do I show $\lim_{x\to\infty}f(x) = \lim_{x\to\infty} f '(x)=0$ if $\lim_{x\to\infty}f '(x)^2 + f(x)^3 = 0$? [duplicate]

$f(x)$ is a real valued function on the reals, and has a continuous derivative such that $$\lim_{x\to\infty} f'(x)^2 + f(x)^3 = 0.$$ How do i show that $$\lim_{x\to\infty} f(x) = \lim_{x\to\infty} ...
1
vote
1answer
17 views

Problem regarding polynomials and partial derivatives

Let $P:\mathbb{R}^n\rightarrow\mathbb{R}$ be the homogeneous polynomial of degree $k$: $$P(x)=\sum_{|a|=k}c_{\alpha}x^{\alpha}$$ How can I show: $\partial^{\beta}P(x)=\beta !c_{\beta}$ for all ...
0
votes
0answers
31 views

In the derivative of an integral, can I compare the size of the limit effect to the area effect?

I have minimal formal math training and sometimes encounter problems like this where I am not sure what relevant techniques are available to use. Thanks for any advice you can give. My integral looks ...
0
votes
0answers
27 views

Finite difference expressions for spherical coordinates

I have information in a spherical grid, that is, I have a value for each combination of $\rho$, $\psi$, $\theta$ ($\rho$ - distance from origin to center of grid cell, $\psi$ - elevation angle to ...
2
votes
4answers
75 views

Is this correct? $ {d \over dy} (1+xy)^y = (1+xy)^y \cdot (1+x \cdot \ln(1+xy))$

I know the formula $ {d \over dx} x^x = x^x \cdot( 1+ \ln x ) $, but is below evaluation correct? $ {d \over dy} (1+xy)^y = (1+xy)^y \cdot (1+x \cdot \ln(1+xy))$
1
vote
1answer
68 views

A mean value theorem involving two functions [duplicate]

Let $f,g:[a,b] \rightarrow \mathbb{R}$ be continuous in $[a,b]$ and differentiable in $(a,b)$. Prove that there is a point $c \in (a,b)$ such that: $$[f(b)-f(a)]g'(c) = [g(b)-g(a)]f'(c).$$ I ...
-2
votes
3answers
42 views

Differentiation with respect to a constant variable? [closed]

Let $y=f(x)$. If we are trying to find $f^{\prime}(x)$ and we know that in the domain we are trying to find $f^{\prime}(x)$ in, $x$ is constant , then what is $f^{\prime}(x)$? Is it zero?
2
votes
1answer
65 views

What did I do wrong?

So, I have found the following problem. This problem is a multiple-choice one, and I have to pick the correct answer. The problem, gives a function $f:D \to R$, $$f(x)=\frac{xe^x}{e^x-a}$$ with $a$ ...
4
votes
5answers
147 views

Differentiate expression involving reciprocal of square roots.

I need to differentiate $$5\over 2+\sqrt{1+3x}$$ I can get the answer from Wolfram Alpha but I'm trying to understand the working. Do I use the chain rule? My calculus is at the basic level.
2
votes
1answer
32 views

Find all $n \in \mathbb N$ such that $g(x) = 100|x+1| - \sum_{k=1}^{n}|x^k+1|$ is differentiable $\forall x$

Find all $n \in \mathbb N$ such that $$g(x) = 100|x+1| - \sum_{k=1}^{n}|x^k+1|$$ is differentiable $\forall x$. It's my high school calculus problem. Is it possible to solve this problem in the high ...
2
votes
1answer
30 views

Example where partial derivatives commute but are not continuous.

I am looking for an example of a function $f:\mathbb R^2\to\mathbb R$ such that there is a point $x\in\mathbb R^2$ with the following properties: 1) All partial derivatives of second order exist in a ...
2
votes
2answers
28 views

Necessary condition for local maximum

Let $\Omega\subset \mathbb{R}^n$ open, bounded and let $f:\Omega\to\mathbb{R}$ be a $C^2$-function. I want to prove: Necessary for a interior maximum $x_0\in\Omega$ is that $D^2f(x_0)$ is negative ...
2
votes
1answer
44 views

Calculate the derivative

I'm asked to find the derivative of the following: $$ \sqrt[4]{x} + \sqrt[3]{3x} $$ I attempted to solve the problem and got the following result, but my book says I am wrong. $$ \frac 14x^{-\frac ...
4
votes
4answers
232 views

Is My Proof that $\pi^e < e^{\pi}$ Valid? [duplicate]

The other day, a math teacher at my college gave me a challenge problem: Prove that $$\pi^e < e^{\pi}$$ without using a calculator. The next day, I found a valid proof, but I used a log table ...
0
votes
2answers
64 views

Differentiate the Function: $g(u)=\ln\left(\frac{\ln\ u}{1+\ln\ (2u)}\right)$

$$g(u)=\ln\left(\frac{\ln\ u}{1+\ln\ (2u)}\right)$$ $$=\ln\ (\ln\ u)-\ln(1+\ln\ (2u))$$ This is the part where I get a little confused. Keep in mind I am using this formula $$\frac{d}{dx}[\ln ...
0
votes
2answers
52 views

Measure of curve smoothness

Could someone please give me the intuition behind using integral of squared second derivative as a measure of curve smoothness? I was thinking that since curvature measures how fast a curve changes, ...
1
vote
2answers
55 views

Differentiate the Function $ h(z)=\ln\sqrt{\frac{a^2-z^2}{a^2+z^2}}$

Differentiate the function $$h(z)=\ln\sqrt{\frac{a^2-z^2}{a^2+z^2}}$$ My try: $$h(z) = \frac{1}{2}\ln\left(a^2-z^2\right)-\frac{1}{2}\ln\left(a^2+z^2\right)$$ so $$h'(z) = ...