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

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3answers
23 views

Related Rates Shadow Problem

The question is as follows: A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. When he is 10 feet from the base of the light, (a) at what rate ...
0
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1answer
27 views

Second derivative with implicit differentiation

Question: Determine whether the given relation is an implicit solution to the give differential equation. Assume that the relationship does define y implicitly as a function of x and use implicit ...
0
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0answers
22 views

how can I prove that a derivative of an implicit function is bounded?

I have the following implicit function $V(\tau,\mu)$. The function is bounded and continuous and differentiable on $\mathbb{R}$. What other properties or assumptions should I make or what conditions ...
4
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3answers
75 views

Implicit Differentiation. Please help me understand why!

I am trying to understand implicit differentiation; I understand what to do (that is no problem), but why I do it is another story. For example: $$3y^2=5x^3 $$ I understand that, if I take the ...
6
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2answers
140 views

Understanding implicit differentiation with concepts like “function” and “lambda abstraction.”

In high school, we learned to reason like so: $$(*) \qquad \frac{d}{dx}(x^2+x) = \frac{d}{dx}(x^2)+\frac{d}{dx}(x) = 2x+1$$ Now that I know more, I can "reanalyze" this chain of reasoning using ...
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1answer
29 views

Differentiation of an implicit function

$$ x = y^x $$ been working through some implicit functions and this has me a bit, I don't think my answer is correct I have treated $y$ as $y(x)^x$ and differentiated with respect to x
4
votes
3answers
270 views

Angle between two parabolas

I'm a little confused about a problem that asks me to find the angle between the two parabolas $$y^2=2px-p^2$$ and $$y^2=p^2-2px$$ at their intersection. I used implicit differentiation to find the ...
0
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1answer
24 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 ...
3
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2answers
82 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
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1answer
27 views

Differentiation Derivation - Limits Question [duplicate]

Hi, I encountered these perplexing questions in my study of the derivation of trigonometric differentiation. Could someone help?
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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
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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
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3answers
35 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
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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
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2answers
66 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
76 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
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3answers
33 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
71 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
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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
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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 ...
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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, ...
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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
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1answer
32 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
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2answers
32 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
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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
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1answer
30 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
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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
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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
28 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
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2answers
65 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
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2answers
42 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
119 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
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2answers
131 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
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3answers
71 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!
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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
39 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
66 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
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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
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1answer
29 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
112 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
132 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
15 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
48 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: ...