For questions on finding and evaluating derivatives when a function is defined implicitly.

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0
votes
1answer
22 views

Finding a tangent line with implicit differentiatio

Find the tangent line on point $P$ for this curve $(x + 2)^2 + (y - 3)^2 = 37$ on $P(4,4)$ I tried implicit differentiating $2(x + 2) + 2(y - 3)y' = 0$ I'm not sure if solving for $y'$ is the ...
2
votes
2answers
65 views

How do I differentiate the following implicit function

$$\sqrt{x^2+y^2} = e^{\sin^{-1} \left( \frac{y}{\sqrt{x^2 + y^2}} \right) } \\ \text{find} \quad \frac{d^2x}{dy^2} \quad \text{i.e. prove} \\ \frac{d^2x}{dy^2} = \frac{2 \left( x^2 + y^2 \right) ...
-1
votes
1answer
23 views

Differentiation Derivation - Limits Question [duplicate]

Hi, I encountered these perplexing questions in my study of the derivation of trigonometric differentiation. Could someone help?
0
votes
1answer
25 views

Derivatives of Implicit Functions (Abstract Case)

I have never been good at differentiation of implicit functions in cases when in a function is given, much less in abstract cases with composite functions. Hopefully someone can help me get started on ...
3
votes
2answers
48 views

Enlighten me… the science behind differentiation [duplicate]

This a tricky math question I encountered. I know a little bit about the answer. But I want somebody who is very good at math to help me find the real reason behind this. OK Lets start $1^2 = 1$ ...
3
votes
3answers
31 views

Differentiate $y^2-2xy+3y=7x$ w.r.t. $x$. Hence show that $\frac{d^2y}{dx^2}(2y-2x+3)=\frac{dy}{dx}(4-2\frac{dy}{dx}).$

Differentiate $y^2-2xy+3y=7x$ w.r.t. $x$. Hence show that $\frac{d^2y}{dx^2}(2y-2x+3)=\frac{dy}{dx}(4-2\frac{dy}{dx}).$ I differentiated $y^2-2xy+3y=7x$ w.r.t. $x$ and got: ...
2
votes
2answers
36 views

find $\frac{dy}{dx}$ in terms of $x$ and $y$ of $x^2y^2=\frac{(y+1)}{(x+1)}$ - basic question

Find $\frac{dy}{dx}$ in terms of $x$ and $y$ of $x^2y^2=\frac{(y+1)}{(x+1)}$ Ok so using the product rule on the LHS and the quotient rule on the RHS I differentiated both sides of the equation ...
1
vote
2answers
65 views

A curve has equation $\arctan(x^2)+\arctan(y^2)=\pi/4$ [closed]

A curve has equation $$\arctan(x^2)+\arctan(y^2)=\frac{\pi}4$$ a) Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$. b) Find the gradient of the curve at the point where $x=\frac{1}{\sqrt{2}}$ and ...
0
votes
1answer
55 views

Hessian of composite function

Let $\cal{J}(y) : \mathbb{R}^n \rightarrow \mathbb{R}$. Furthermore, suppose $y := h(x)$ with the nonlinear mapping $h : \mathbb{R}^n \rightarrow \mathbb{R}^n$ twice differentiable. The Jacobian of ...
1
vote
3answers
32 views

Write down the equation of the tangent to $x^2-3y^2=4y$ at the point $(x_1,y_1)$.

Write down the equation of the tangent to $x^2-3y^2=4y$ at the point $(x_1,y_1)$. The textbook gives the answer as $xx_1-3yy_1=2(y+y_1)$ and I'm not sure how it got there. Okay so I ...
5
votes
4answers
70 views

Find $\frac{d^2y}{dx^2}$ as a function of $x$ if $\sin y+\cos y=x$

Find $\frac{d^2y}{dx^2}$ as a function of $x$ if $\sin y+\cos y=x$ Ok bit confused as my textbook gives the answer to this problem as: $$\frac{d^2y}{dx^2}=\pm\frac{x}{\sqrt{(2-x^2)^3}}$$ So I ...
0
votes
1answer
17 views

Trouble following simplification stage of implicit differentiation problem - basic question

Show that the equation of the tangent at the point $(x_1,y_1)$ to the curve with equation $x^2-2y^2-6y = 0$ is $xx_1-2yy_1-3(y+y_1)=0$ Alright I'm having a problem following an example from my ...
0
votes
0answers
12 views

Derivative of implicit function - possible to bring in specific form?

Let $f(\alpha) := \sum_{j=0}^{N-1}\alpha^j = \frac{1-\alpha^N}{1-\alpha}$. I am analyzing an implicit equation of the form $g(v,\alpha) := f(\alpha) - \frac{c}{v} = 0$, where $c$ is a positive ...
0
votes
2answers
80 views

How to differentiate $y$ defined by the equation $\sin(x+y) =y^2 \cos x$?

Given: $$\sin(x+y) = y^2 \cos x$$ find $dy/dx$. $$\cos(x+y)(1+y')= \text{...product rule...}$$ how do we get the left one? I am looking at the solution. I tried replacing $\sin(x+y)$ with $\sin x ...
1
vote
2answers
44 views

What is the derivative of $\log_x(A)$ where $x$ is the base (differentiaition with respect to $x$)

I want to find out what $\frac{d}{dx}\log_x A$ is? I did this so far but I'm not sure. $y = \log_x A \Longrightarrow x^y = A$ so, $d/dx(x^y) = d/dx(A)$ [differentiating both sides w.r.t $x$] then, ...
0
votes
0answers
23 views

$AX=B$: How to solve for $X$ if elements of matrix A are matrices

Objective: I am trying to solve for $C$ in 2D space (x,y) and time from following PDE. $$\begin{align} \text{PDE: }\frac{\partial C}{\partial t} + \nabla\left(v.C - D\nabla{C} \right)= \alpha.C ...
1
vote
1answer
29 views

Is this related rates equation set up correctly?

This is the question: Water is being poured into a tank at $10ft^3/min$. Find the rate at which the water level is increasing when the depth of the water is ${1\over 3}ft$ Here is a picture of the ...
1
vote
1answer
20 views

Implicit function theorem conclusion notation?

I am working through implicit function theorem for the first time, and I have the following understanding. Given a system of $n$ equations, \begin{equation} f_i(x_1,\dots ,x_m,y_1,\dots , y_n)=0,\ \ \ ...
0
votes
2answers
26 views

How do I solve this related rates problem using implicit differentiation?

This is the question: Two cars leave from the same point at the same time. One car travels north at a rate of 60 miles per hour and the other travels east at a speed of 80 miles per hour. How fast is ...
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
1answer
29 views

Implicit differentiation and linear approximations

Consider the implicit function $$(w(x)+1)e^{w(x)}=x.$$ I need to approximate $w(1.1)$ using the fact that $w(1)=0$. Could you give me any hints?
0
votes
1answer
41 views

Implicit equation. Can it be solved?

Is it possible to find a function $x:[0,T]\to [0,x_0]$ such that, for a fixed $0<\lambda<1$ we have: $$\dfrac{1}{1+\lambda}\left (1-\dfrac{x(t)}{x_0}\right )^{1+\lambda} +\dfrac{1}{1-\lambda} ...
2
votes
1answer
56 views

Did I do this implicit differentation right? [closed]

I have just solved an implicit differentiation question and feel that I have made a mistake after checking some online calculators. The questions states to use implicit differentiation to find dy/dx ...
2
votes
2answers
27 views

How do I take the implicit partial derivative when the variable is not equal to the equation?

So my homework problem is as follows: $$ \text{Consider the equation }xz^2-6yz+4log(z)=-1 \text{ as defining }z\text{ implicitly as a function of } x \text{ and } y \text{.}\\ \text{The values of } ...
0
votes
2answers
37 views

Finding the equation of parabolas with axis parallel to the x-axis

I've seen a post like this but it's on hold and doesn't really help, can someone give me a hint or a step by step solution on how to solve this? I think the other guy is from another section of my ...
0
votes
2answers
41 views

Implicit Differentiation of $\sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} = 6$ [closed]

Find $\frac{dy}{dx}$ when $\sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} = 6$.
0
votes
1answer
91 views

Find the differential equation of all circles of radius 1 and centers on $y=x$

Find the differential equation of all circles of radius 1 and centers on $y=x$, I've answered several problems with circles finding its equation but not like $y=x$ can someone please explain this to ...
1
vote
2answers
108 views

Find the differential equation of all tangent lines of parabola $y^2=4x$

My professor said that it's $x(y')^2-yy'+1=0$ but how? I drew it and I think it open to the right $90^\circ$ but I can find the solution to differentiate
1
vote
3answers
70 views

The slope of the tangent to the curve $10x^3+5x^2y+4xy^2+6y^3=25$

I am having problems understanding how to solve this equation: The slope of the tangent to the curve: $$10x^3+5x^2y+4xy^2+6y^3=25$$ at the point $(1,1)$. Any help would be appreciated, thanks!
-2
votes
3answers
45 views

Differentiation using logarithms.

the variables $x$ and $y$ are positive and related by $$x^a\cdot y^b=(x+y)^{(a+b)}$$ where $a$ and $b$ are positive constants. By taking logarithms of both sides, show that ...
0
votes
1answer
38 views

Implicit function theorem of ${ x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }=yf\left( \frac { z }{ y } \right) $

If $z=z(x,y)$ This implicitly given by ${ x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }=yf\left( \frac { z }{ y } \right) $ I need to prove $({ x }^{ 2 }{ -y }^{ 2 }{ -z }^{ 2 })\frac { \partial z }{ \partial ...
2
votes
5answers
65 views

for each $x>1 , \frac{x-1}{x}\ < \ln x < x-1$

I tried to prove this with differentiation: when $x >$ 1, all 3 functions are positive and when $x = 1$, all 3 reaches zero. And the derivatives are varying like ...
0
votes
1answer
36 views

n-th derivative with respect to $\frac{1}{x}$

Is it any easy way to calculate : $\frac{d^n x}{d\left(\frac{1}{x}\right)^n}$ for arbitrary $n\in\mathbb{N}$ ? (for $n=1$ it is obvious, but for $n>1$ the formula for $n$-th derivative of ...
1
vote
1answer
52 views

Alternate formula for $Γ(t + 1)$

I believe that: $$\frac{d^{n}}{dx^{n}}[x^{n}] \equiv Γ(n + 1) \equiv n!$$ Would this have any application, if it has not already been discovered, which I am almost certain that it has?
0
votes
1answer
34 views

Differentiating a sum involving logs

I was doing the problem provided in the picture but I do not understand how do they obtain the answer. I am not sure how to differentiate the sum. I end up getting: alpha - 1 - 1/K. I believe I need ...
0
votes
0answers
17 views

How can we define regular curves implicitly?

Let $F:D\subseteq\mathbb{R}^2\to\mathbb{R}$, $D$ open and connected set, be a $C^1 (D)$ application. What are the minimum requirements for $F$ such that the solutions of the equation $F(x,y)=0$ are ...
2
votes
1answer
27 views

Difficulty Understanding Implicit Differentiation

I am struggling with an Implicit Differentiation question which is as follows: $z = (7x^4)*\ln(x)4$ where $z$ and $x$ are functions of $t$. $\frac{dx}{dt} = 4$ when $x = e$. Calculate ...
3
votes
1answer
110 views

Clarification on Implicit Derivatives steps

I have been attempting to wrap my head around this problem for a couple days now. I've attempted numerous different iterations to try and find how the answer is derived, but I just don't see the ...
3
votes
3answers
51 views

Logarithmic Differentiation equation, Help!

So, I have to differentiate this via $\log$. I am still learning, so please be patient, I will try to explain everything I did. Please tell me if it is correct. ...
2
votes
1answer
131 views

$y^5 =(x+2)^4+(e^x)(ln y)−15$ finding $\frac{dy}{dx}$ at $(0,1)$

Unsure what to do regarding the $y^5$. Should I convert it to a $y$= function and take the $5$ root of the other side. Then differentiate? Any help would be great thanks.
1
vote
1answer
14 views

Linearization of an implicitly defined function

$f(x,y,z)=e^{xz}y^2+\sin(y)\cos(z)+x^2 z$ Find equation of tangent plane at $(0,\pi,0)$ and use it to approximate $f(0.1,\pi,0.1)$. Find equation of normal to tangent plane. My attempt: I found that ...
0
votes
2answers
47 views

Using implicit differentiation with a fraction

How do I solve this? What steps? I have been beating my head into the wall all evening. $$ x^2 + y^2 = \frac{x}{y} + 4 $$
0
votes
1answer
33 views

why does $\frac{d}{dx} log_b(x)$ not = $\frac{lnb}{x}$?

I know that $log_b(x) = \frac{lnx}{lnb}$, and that differentiating $$\frac{d}{dx}(\frac{lnx}{lnb}) = \frac{1}{lnb}\frac{d}{dx}(lnx)=\frac{1}{xlnb}$$, so where is my mistake when I do it this way: ...
3
votes
1answer
48 views

Use implicit differentiation to find an equation of the tangent line to the curve at the given point

Use implicit differentiation to find an equation of the tangent line to the curve $$x^2+xy+y^2=1$$ at $(1,1)$. I am not sure how I should work this out because the given point is not on the ...
0
votes
0answers
42 views

How do I write this equation as a tridiagonal matrix to write the $n+1$ implicit formula?

I am doing a homework problem for my Applied Numerical Methods class, and I've worked the problem up to this point: $$ \large \frac{u_m^{n+1} - u_m^n}{k}=\frac{u_{m+1}^{n+1} - 2u_{m}^{n+1} + ...
0
votes
3answers
68 views

Struggling with integration/differentiation

quick question as I'm sure this is simple but it has me stumped. I have to integrate and differentiate this equation. Not sure on the exponential, had a couple of goes but it doesn't look right. ...
-3
votes
1answer
59 views

Confused by partial derivatives

Right, I have a question here, about the following: Use implicit differentiation to find the first and 2nd derivative. $$x^{3/5}+y^{3/5} = 7$$ The answer is: $$\begin{align}\frac{dy}{dx} &= ...
2
votes
1answer
60 views

Find $dy/dx$ of $(xy^2)+5 = x + 2y^2$

For the solution I got $$\frac{y^2-1}{ 4y-2xy} = dy/dx$$ I just want to know if this is correct. Also it says to evaluate $dy/dx$ at $(1,2)$. Would the solution to that be $3/4$?
0
votes
1answer
41 views

Related Rates problem, xy=4

$xy=4$ $a)$ Find $\dfrac{\mathrm dy}{\mathrm dt}$ when $x=8$, Given $\dfrac{\mathrm dx}{\mathrm dt} = 10$ $b)$ Find $\dfrac{\mathrm dx}{\mathrm dt}$ when $x=1$, Given $\dfrac{\mathrm ...
2
votes
0answers
29 views

Tangent line of a lemniscate at (0,0)

I need to find the tangent line of the function $y=g(x)$ implicitly defined by $(x^2+y^2)^2-2a^2(x^2-y^2)=0$ at $(0,0)$, but I don't know how. I can't use implicit differentiation and evaluate at ...