0
votes
0answers
26 views

True or false: differentiation. [on hold]

If the function $f(x,y): \mathbb{R}^2 \longrightarrow \mathbb{R}^3$ is differentiable at $(2,-1)$ with a tangent plane such as $z= 2x - 3y + 2$, then the function $g(x,y)= 3x - 2f(x,y) + 5$ is ...
3
votes
2answers
134 views

Real analysis question involving a linear ODE

Where do I start with this one? This question is really quite difficult..
1
vote
0answers
32 views

Total differentiation

For each of the functions below use the total diferential to approximate the change in $Y$ due to the given changes in $X$ and $Z$: $Y= X^2 + 4X -Z^2 -2XZ$, where $X=1$ and $Z = 4$ , and $\Delta X=2$ ...
20
votes
2answers
297 views

When does $(uv)'=u'v'?$ [duplicate]

In any calculus course, one of the first thing we learn is that $(uv)'=u'v+v'u$ rather than the what I've written in the title. This got me wondering: when is this dream product rule true? There are ...
1
vote
0answers
40 views

Show that $y/x$ tends to a finite limit as $x \to + \infty$ and determine this limit.

Let $y=f(x)$ be that solution of the differential equation $$y' = \frac{2y^2+x}{3y^2+5}$$ which satisfies the initial condition $f(0)=0$. (Do not attempt to solve this differential equation.) (a) ...
1
vote
2answers
39 views

Help with separable differential equation? $\frac{dy}{dx} =2y^2$

I'm new to separable differential equations, and I'm stuck on this question: $\frac{dy}{dx} =2y^2$ Using the initial condition $y(2)=3$, find $y(1)$. So far I've integrated to get $\frac{dy}{dx} ...
0
votes
0answers
15 views

How to find the solution in $x$ when the differential equation is in $\partial_y$ such that $\partial_y = \partial_x + f(x)$

For my problem, the differential equation can be simplified by defining a new operator like $\partial_y = \partial_x + f(x)$. But at the end I need a solution in terms $x$. How can I do that? As an ...
1
vote
1answer
32 views

second order nonhomogeneous differential equation help? (easy)

finn the general solution to the nonhomogeneous differential equation $$y''+ 2y'-3y = 5e^{-3x}$$ and I have to use undetermined coefficients? ok so what I did was found out that the homogeneous ...
0
votes
3answers
31 views

How to simplify a complicated partial differentiation

Given $u = (t^{-1/2})\exp\left[{\frac{-x^2}{4k^2t}}\right]$, what is the best way to differentiate $u$ with respect to $t$, as well as $u$ with respect to $x$? I am having a very difficult time trying ...
1
vote
1answer
51 views

Finding the derivatives of inverse functions at given point of c

Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post ...
0
votes
1answer
33 views

Differential equation, symmetric about 0?

Solving the following numerically (with different values of $u(-1)$) $(2-\cos(\pi x))u''(t) + u(t) = 1$ and $u(-1) = u(1)$ the solutions seem to be symmetric about $0$. Is it true in general (ie no ...
0
votes
1answer
31 views

Find all solutions to a particular differential equation

Find all solutions on ${R}$ of the differential equation $ y' = 3|y|^ \frac{2}{3} $ I believe I need to use separation of variables, but it will only work if the initial condition is nonzero. ...
1
vote
1answer
22 views

simple derivative question

What is the derivative of $x''(t) = \cos(tx(t))$ with respect to $x'(t)$? I am at a loss at what to do for this simple stated problem. I am not exactly sure what it means to take the derivative with ...
2
votes
1answer
75 views

Solve the differential equation: $(y^2-xy)dx+x^2dy=0$

$$(y^2-xy)dx+x^2dy=0$$ Need step by step answer. Thank you in advance for your help
3
votes
2answers
78 views

Finding the second derivative; What am I doing wrong?

Original Question: $xy+y-x=1$ Find the second derivative; $d^2y\over{dx^2}$$(xy+y-x=1)$ We are allowed to use either notation as far as I know: ${dy\over{dx}}$ or ${y'}$. Because ...
2
votes
2answers
94 views

Solve the following differential equation: $xy' - y = x^2$

I'm preparing to exam in Linear Algebra $2$ and I have problems with differential equations.. For example, the following exercise: Solve the following differential equation: $xy' - y = x^2$. I ...
0
votes
1answer
44 views

Second Order Differential Equation Question

Got this question on my FP3 homework - if anyone could help me out I'd really appreciate it. .
0
votes
0answers
39 views

Derivation of differential equations for model with input and output derivatives

Consider the following model: Where the top most spring is a non-linear one with the following characteristic: $F_s=k_{s,1}*x+k_{s,2}*x^3$ The middle damper and spring are both linear ones. The ...
1
vote
2answers
118 views

Implicit form of general equation

Find, in implicit form, the general solution of the differential equation: $$\frac{dy}{dx}= \frac{2y^4e^{2x}}{3\left(e^{2x}+7\right)^2}$$ I am struggling to make any sense of this. What I have ...
1
vote
1answer
13 views

Assuming F(x=) # of people living within X radius of walmart is F' nonpositive or nonnegative

So F(x)=# of people living within a radius of x miles from walmart First question - What does F(3) represent - my answer - 3 people within 3 miles from walmart second question - What does quanity ...
0
votes
2answers
61 views

How does expanding by Taylor's theorem work here?

The problem I am trying to figure out a step in the proof of this book (p. 245), where it goes like this: \begin{equation}\tag{a}\label{eq:equal} F_i(x^0; t + dt) = F_i(x; dt)\end{equation} ...
3
votes
2answers
44 views

Derivative of a polynomial

First of all, I would like to say I'm new to Mathematics StackExchange, so pardon me if there're any mistakes (until I read the right formatting rules!). That said, we are currently learning ...
1
vote
1answer
37 views

Properties of a derivative of a function $f(x)$ expressed as a function of $f$

Consider a differentiable function $f(x)$ and let $u(f(x)) = f'(x)$. Normally we solve for $u$ as a function of $x$, but we can also express it as a function of $f$. Some examples: If $f(x) = ...
2
votes
1answer
74 views

Derivative of a Linear Map

I'm devastatingly incompetent at linear algebra and multivariable calculus. I just cannot understand it at all. Here's the easiest problem from my homework, and my attempt at solving it, and where I ...
1
vote
2answers
33 views

Prove that the average of $D_wD_wf(x_0,y_0)$ over all unit vectors $w$ is equal to $\frac{1}{2} \Delta f(x_0,y_0)$ for any smooth function $f$.

Here is a challenge problem from my math professor: Let $w$ be a unit vector in $\mathbb{R}^2$, and let $D_w$ denote the directional derivative with respect to $w$. Prove that for any smooth ...
1
vote
1answer
31 views

Taylor series expansion of $f(x)=\frac{sinx}{x-\pi}$ at $x=\pi$

I was solving it and on one step I need to find the 2nd Derivative of $f(x)$, I am getting -1/3, but according to book it's -1/6.Please help me out here.
2
votes
2answers
50 views

Suppose $f$ is real-valued function defined on $[1,\infty)$ with $f(1)=1$.Suppose, moreover, that $f$ satisfies…

Suppose $f$ is real-valued function defined on $[1,\infty)$ with $f(1)=1$.Suppose, moreover, that $f$ satisfies $$f'(x)=\dfrac{1}{x^2+f^2(x)}$$. Show that $f(x)\le 1+\pi/4$ for every $x\ge 1$. My ...
0
votes
0answers
100 views

Question on derivative

If $F_2(u)=\frac12 (Au,u)$ where $A$ is a continuous and self adjoint operator and $\eta$ the flow défini l'o.d.e $$ \begin{cases} \displaystyle\eta '(s)=- \frac{A\eta(s)}{||A\eta (s)||}\\ \eta(0)=u ...
2
votes
2answers
80 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 ...
0
votes
1answer
23 views

derivation of a differential Eq

Look at $F(u) = \frac{\partial u}{\partial t}-\nabla \cdot (a(u)\nabla u)$. My question is, what $F'(u)$ is. I need this for the linearization of a PDE. The idea is to use the newton-approximation. ...
0
votes
1answer
32 views

Elementary differentiation question on derivation of p.d.f. of function of random variable

Let $G(y) = \Pr(Y \le y) = 1 - F(\frac{1}{y})$. Then apply the chain rule (assuming $y \ne 0$ and $F(x)$ is differentiable at $x = 1/y$) and we have $$g(y) = \frac {d\ G(y)}{dy} = \frac{-d\ F(x)}{dx} ...
1
vote
1answer
53 views

How to make this change of variables?

"Show that if we introduce the independent variable $x = \sqrt{\frac{z}{L}}$ then the equation $zZ''(z) + Z'(z) + v^2Z(z)=0$ becomes $Z''(x) + \frac{1}{x} Z'(x) +4v^2LZ(x)=0$ for $0<x<1$. So ...
0
votes
2answers
248 views

Clarification on Rules Differentiation and First Principles Derivatives

My grade 11 class has just started differential calculus, the one area seemingly glazed over in our book. We have covered some simple rules of differentiation, like f(x) = n x^(n-1), and have applied ...
0
votes
1answer
222 views

Finding dy/dx as a function of x for a dog-walker dragged by a dog travelling in a straight line

Hello. I was wondering if anyone could provide some insight into how to solve the following Calculus word problem: Max is walking his dog Beau in the Cartesian plane, with the leash between them at ...
-2
votes
1answer
31 views

Differentiate the functions trigonometry and derivatives

Differentiate $ln(\sec x+ \tan x)$ and $ (\sin x)^3\cos 3x+ (\cos x)^3\sin 3x$ with respect to $x$, simplifying where possible. Find the first and second derivatives (with respect to x) of the ...
0
votes
2answers
30 views

Finding Fourier transform of initial condition

Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $$u(x,0)=f(x).$$ Let ...
1
vote
1answer
76 views

A nonlinear first-order differential equation

How do we solve the following differential equation? $$(y{}')^{2}+p(x)(1+y^2)^{3}=0$$
0
votes
1answer
37 views

numeric differentiate: show that the relative mistake can be at 100%

i have $f(x) = x+1$, a physical size, and the values $\tilde{f}(x_i)$ are measured at equally spaced points $$x_i=ih, \qquad 0 \leq i\leq 10^3, \qquad h=10^{-3},$$ with a maximum relative mistake of ...
2
votes
1answer
68 views

Understanding how to take derivatives with matrices

Currently we are doing 2nd order differential equations (we already did systems of homogenous two first order equations) and now that we have non-homogenous 2nd order equations we are doing method of ...
0
votes
2answers
83 views

basic differential question

I need guidance on this problem. Could someone lead me in a direction of how I should go about doing this question. Is there some sort of proof involved in this question? No need to solve the question ...
3
votes
1answer
60 views

Calculus Proof Question (Differentiable)

Prove that if f is a differentiable odd function then f ' is an even function. Hello, I don't completely understand what this question means? What does it mean ...
1
vote
0answers
83 views

How to show that the first derivative is bounded in a function

\begin{equation} y=\arccos\left( -\frac{1}{2\left(Dr^{\dfrac {|\sin(2x+\theta)|}{M\sin x\sqrt{A+2B\cos(2x+\theta)}}}+1\right)} \right) \nonumber \end{equation} How to show in above function the ...
1
vote
2answers
88 views

Derivative of mixed matrix terms with inverse matrix

I've been trying to solve two matrix derivative terms including an inverse matrix but I am unable to find a clue : 1) Derivative of $KG^{-1}J$ with respect to $G$. 2) Derivative of ...
0
votes
2answers
137 views

Determine an equation for the tangent to the graph of f(x) at point P.

Determine an equation for the tangent to the graph of f(x) at point P. Use of CAS tool allowed. a) f(x)= 3/(1+√x) , P(4, 1) ...
1
vote
1answer
46 views

Differential Proving Question (Calculus)

If f satisfies |f(x)|<=|x|^9 for all x , prove that f is differentiable at 0. Hello I am having trouble understanding this question, could someone explain what this question is asking? ...
1
vote
1answer
271 views

Proof of Chain Rule using Nonstandard Analysis

I am trying to make an introduction and make myself comfortable with the Nonstandard Analysis in order to gain intuition for derivatives and integration. I am trying to prove myself the famous Chain ...
2
votes
1answer
37 views

How to differentiate an integration?

$$\int_{q_1}^{q_2}f_T(t)dt=0.6826\ldots(1)$$ How differentiating equation $(1)$ with respect to $q_1$ yields $$f_T(q_2)\frac{dq_2}{dq_1}-f_T(q_1)=0$$
1
vote
1answer
34 views

Issues in calculating the gradient

I am trying to calculate the gradient of a certain expression. I am not sure if it's possible. I have the following $f(\alpha_1,\alpha_2,\Lambda) = \log(|2Q_1+2Q_2 +2Q_3|)$ $Q_1$ is a diagonal ...
1
vote
0answers
50 views

Confusion related to the gradient of the sum of a smooth and non-smooth function

I have this confusion related to gradient. Let my function $f(x) = g(x) + h(x)$ where $g(x)$ is a differentiable function and $h(x)= \lambda\|x\|_1$ $g(x)$ is differentiable but $\|x\|_1$ is not at ...
1
vote
1answer
45 views

how to prove a inequality

I have to prove that if I have $$x'=\sin \left [ 2(x+t) \right ]$$ and $x(t), y(t)$ are two maximal solutions, then $$|x(t)-y(t)|\leq |x(t_0) -y(t_0)|e^{2|t-t_0|}$$ I have thought about prove that ...