Questions tagged [implicit-differentiation]

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

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Hypocycloid problem. Show that the portion of every tangent line in the first quadrant is equal 1.

Problem statement: Show that every tangent line to the curve $x^{2/3} + y^{2/3} = 1$ in the first quadrant has the property that portion of the line in the first quadrant has length 1. The textbook ...
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Monotonicity of implicitly defined function

Let $f(x,y):\mathbb{R}^2\rightarrow\mathbb{R}$ and $g(y):\mathbb{R}^2\rightarrow\mathbb{R}$ be $C^2$-differentiable functions. Let $f(x,y)$ and $g(y)$ be strictly decreasing in $y$, and let $f(x,y)$ ...
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What are the conditions for a function to be a unique implicit function?

My question is pertaining to part b. What are the conditions for a function to be a unique implicit function? Do we only have to check if the partial derivative at (x0, y0) evaluate to a number other ...
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What is the slope function in terms of $x$ of the Folium of Descartes' function?

I wish to have a $2D$ function that, when plot in a cartesian plane of coordinates, it draws a curve that returns the slope of the Folium of Descartes' function for any given $x$ coordinate as $y$ ...
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How would I use implicit differentiation to find an equation of the tangent line to the curve at the given point.

So the question is: Use implicit differentiation to find an equation of the tangent line to the curve at the given point. $$x^2 + y^2 = (5x^2 + 4y^2 − x)^2,\text{ at } (0, 1/4) \text{ (cardioid)}$$ ...
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1 answer
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How do I use implicit differentiation to find an equation of the tangent line to the curve at the given point. [closed]

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. tan(x + y) + sec(x − y) = 2, (𝜋/8, 𝜋/8) I have no Idea how to solve this problem. If anyone is ...
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Determine a curve with equation $x^2+ay^2+bx+cy+d=0$ that has the highest possible order of contact with $y=cos(x)$ in $(0,1)$

Determine a curve with equation $x^2+ay^2+bx+cy+d=0$ that has the highest possible order of contact with $y=cos(x)$ in $(0,1)$ Can somebody help me, I tried to take the derivative of $y = cos(x)$ ...
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1 answer
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If $\frac d{dx} f(x,y)>0$, can I claim that $f(x,y)$ is increasing with respect to $x$?

I have an implicit equation $f(x,y)=0$; computing the derivatives, I see that $\frac d{dx} f(x,y)>0$ while $\frac d{dy} f(x,y)$ maybe positive, or negative. Question. Is this data sufficient to ...
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Implicit differentiation: under what conditions can implicit differentiation not be used? is there a way too tell before solving?

My calculus I book states "in the examples and exercises of this section it is always assumed that the given equation determines y implicitly as a differentiable function of x so that the method ...
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use implicit differentiation to find the derivative of $(x^2+y^2)^4=6x^2y\,$?

I made up a question to practice implicit differentiation with the relation $(x^2+y^2)^4=6x^2y$. this is my solution: Also I am sorry but I don't know how to write the more complex parts of the ...
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Chapter 12 Spivak Calculus: Spivak's comments about implicit differentiation

There is a passage in the Chapter 12 exercises of Spivak's Calculus (which is a book specific to real-valued functions) that reads as follows: In general, determining on what intervals a ...
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Change of variable and Jacobian

I am having a bit of trouble with the following question: Given a region $D$ in the first quadrant bounded by $y = \sqrt{x}$, $y=2\sqrt{x}$, $x^2 + y^2 = 1$ and $x^2 + y^2 = 4$, evaluate: $$\iint_D \...
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If $z=g(x, y)$ is defined by $f(x, y, z)=0$ near the point $(a, b, c)$, find $\frac{\partial g}{\partial x} (a, b)$.

Problem: Suppose $f=f(x, y, z): \mathbb{R}^3\to\mathbb{R}$ is continuously differentiable, $f(a, b, c)=0$ and $\frac{\partial f}{\partial z}(a, b, c)\neq0$. If $z=g(x, y)$ is defined by $f(x, y, z)=0$ ...
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Mistake in the second derivative of implicit function

I have came upon this segment in a paper: I am a bit confused. Is the formula for $\frac{d^2x}{dt^2}$ even true here? Also, it looks like they mistakingly used the notation $\frac{\partial^2 x}{\...
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Using implicit differentiation of multiple varibles

The right-hand side of the equation was easier to differentiate wrt $x$ in which I got: $4x-4y(\frac{dy}{dx})$ However, that was not the case for the left-hand side. The main reason is the term is ...
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Finding a simple differential equation to define an inverse of $\,_2\text F_1(a,b;c;z)$ with respect to $z$ with the Gauss Hypergeometric function.

An “Inverse Gauss Hypergeometric function” with respect to $z$ in terms of a differential equation would define many special case inverse functions. Define: $$\,_2\text F_1(a,b;c;z)=\sum_{n=0}^\infty \...
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Implicit differentiation vs. solve then differentiate

In a calculus assignment, I was asked to find the derivative of the following: $$yx^4=\frac{2}{3}$$ Solving for y then differentiating produces: $$\frac{\partial y}{\partial x}=\frac{-8}{3x^5}$$ The ...
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Differentiate A = $\frac{\sqrt{(π^2r^6 + 9V^2)}}{r}$ with respect to r.

Differentiate A = $\frac{\sqrt{(π^2r^6 + 9V^2)}}{r}$ with respect to r. This function to the best of my knowledge is a multivariable function that can be differentiated partially, so I differentiated ...
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1 answer
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Intuitive explanation of the sign of the derivative of an implicit function.

I have three functions $f(y), g(y)$ and $h(x)$. I know that all three are positive valued and the first is increasing and the last two are decreasing. I also know that $$ g(y)=\frac{f(y)}{h(x)}.$$ The ...
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Partial Implicit Differentiation

I am currently writing a paper related to spotted owl conservation and reading a paper about demographic models for that species. It uses the Euler-Lotka equation, along with some facts about owl ...
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1 answer
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Why does this term appear twice in the step of implicit partial differentiation?

The textbook example gives an implicitly defined function of $$x^3+y^3+z^3+6xyz=1$$ It then walks through the first step of partial differentiation of z with respect to $x$: $$3x^2+3z^2\frac{\partial ...
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2 votes
1 answer
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Maximum present density of matter to avoid Big Crunch

According to currently accepted ideas among astronomers, the universe came into existence about 15 billion years ago in an explosion called the Big Bang. Ever since that time the universe has been ...
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Numerical treatment of radiation boundary condition for 1D implicit heat conduction

I have a question regarding the $\textbf{radiation boundary treatment}$ for the 1D heat conduction equation using the $\textbf{implicit}$ finite difference method. If I use the $\textbf{explicit}$ ...
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1 vote
1 answer
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Limit of implicitely defined function?

I have this implicit equation $F(x, y) = x^2y+e^{x+y} = 0 $. Now this defines a function $y=f(x)$ everywhere except for $x=0$. I need to compute the limit for $x \rightarrow 0^+$. I know that $y = -\...
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Rectangle in a circle of radius a that maximizes x^n+ y^n

Consider a rectangle with sides $2x$ and $2y$ inscribed in a given fixed circle $$x^2 + y^2 = a^2,$$ and let $n$ be a positive number. We wish to find the rectangle that maximizes the quantity $$z = x^...
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1 vote
2 answers
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How fast the water level is rising when the ball is half submerged.

Question: Water is being poured into a hemispherical bowl of radius 3 in at the rate of 1 in3/s. How fast is the water level rising when the water is 1 in deep. In Problem 1, suppose that the bowl ...
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1 answer
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Maximum Total Surface Area due to hole in sphere.

Question My attempt The region between the dotted lines shows the hollow region. The distance MP = $ \sqrt{R^2 - x^2} $ Hollow area created = 2πx×2MP = $4πx\sqrt{R^2- x^2}$ The area removed due to ...
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2 votes
1 answer
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SHORTEST possible time

Here's my solution I don't know what mistake I have done but I always get the same imaginary solution. Can you please solve and check or suggest something?
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1 answer
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Confusion in a step of the derivation of the invariance of Laplace's equation

I'm reading the derivation in Strauss's PDE book. He sets up the following relations: $$ r = \sqrt{x^2 + y^2 +z^2} = \sqrt{s^2 + z^2} , s = \sqrt{x^2 + y^2}, x = s \cos{\phi}, y - s \sin{\phi}, z = r ...
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3 votes
1 answer
102 views

finding derivatives of variables in multivariable taylor polynomial

Given: $$F(x,y) = -6 -4(x-4) +6(y-6) +8(x-4)^2 +9(x-4)(y-6) -4(y-6)^2 + R_2$$ and that $F(4,6)=-6$ find y'(4), y''(4), make sure that the function is applicable for the implicit function theorem. my ...
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0 votes
1 answer
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Implicit differentiation: why I cannot multiple both sides by an expression?

I was solving a problem on implicit differentiation and I was wondering why my answer does not match the given answer. Apparently I multiplied both parts of an equation by an expression to get rid of ...
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-1 votes
3 answers
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Find the value of $d^2y/dx^2$ when $ x=1$ [closed]

If $x^2-x^2y+3xy^2=5$, then find the value of $d^2y/dx^2$ if $x=1$ I have tried changing the subject of the equation to make it y and also tried implicit differentiation but both dont seem to work and ...
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5 votes
2 answers
262 views

Prove derivative of an implicit function

The question: $\sqrt{1+x^2} + \sqrt{1+y^2} = a(x-y)$ prove that $\dfrac{dy}{dx} = \sqrt{\dfrac{1+y^2}{1+x^2}}$ I tried doing it the normal way and got $\dfrac{dy}{dx} = \dfrac{(a\sqrt{1+x^2}-x)(\sqrt{...
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1 vote
1 answer
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Problem in implicit differentiation

Let $$x^2+xy+y^2=2$$ Find $\displaystyle\frac{dy}{dx}$ Applying $\frac{d()}{dx}$ on both sides gives $$\frac{dy}{dx}(2y+x)=-(2x+y)$$ For $x\neq -2y$ $$\frac{dy}{dx}=-\frac{(2x+y)}{(2y+x)}$$ ...
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Solving derivative of $x^x$ without logarithmic differentiation [duplicate]

Been learning calculus, and was taken aback (in a good way) by the challenge of $$\frac{d}{dx}x^x$$ because the rules I've learned so far (chain/product/quotient/implicit differentiation) can handle $...
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2 votes
1 answer
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Construction of derivative as ratio of determinants related to simultaneous system of implicit functions

This is Example C in section 1.1 from Advanced Calculus (2nd ed) by David Widder. $$\begin{cases} v + \log u = xy \\ u + \log v = x - y \end{cases}$$ $$\begin{cases} \frac{1}{u} \frac{\partial u}{\...
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0 answers
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factor derivative

If Dy/Dx = -5sin(5x-3y)+(Dy/Dx(5-3)) Then how can: (1-3*sin(5x-3y))Dy/Dx = -5sin(5x-3y) And so the derivative is -5sin(5x-3y) / (1-3sin(5x-3y)) How can the derivative dissapear , it's factored but ...
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0 answers
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implicit function on connected curves

Suppose $h(x,y)$ is Lipschitz continuous with continuous partial derivatives. Let $$h(0,y) = 0 \quad \forall y$$ and $\gamma : [0,1) \to \mathbb{R}^2$ be a curve beginning from the $y$ axis with $\...
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1 answer
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Apply implicit function theorem on $z^3 - 3xyz = 1$ and $z = z(x,y)$

To solve this I first rearrange $z^3 - 3xyz - 1 = 0$ $\displaystyle\frac{\partial z}{\partial x}$ = $3z^2z'-3xyz'-3yz$ $\displaystyle\frac{\partial z}{\partial y}$ = $3zz' -3xyz'-3z$ $\displaystyle\...
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0 votes
1 answer
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simple q, partial derivative writing formally with function having both implicit and explicit dependence on variable

I have the function $L$ which depends on other variables as well as $n$ explicitly and implicitly as below. $$\frac{\partial L}{\partial n} = \frac{\partial L}{\partial e}\frac{\partial e}{\partial n} ...
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3 votes
3 answers
224 views

Slope of the tangents to the circle $x^2+y^2-2x+4y-20=0$

Find the slope of the tangents to the circle $x^2+y^2-2x+4y-20=0$. After I arranged into a standard form, which is $(x-1)^2 +(y+2)^2=25$ Centre point is $(1,-2)$ radius is $5$ unit. Do I need to do ...
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1 vote
1 answer
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Finding $\frac{\partial z}{\partial x}$ for $x^2z^2\:+\:xy^2\:−\:z^2\:+\:4yz\:−\:5\:=\:0$

How can I find the $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ for $x^2z^2\:+\:xy^2\:−\:z^2\:+\:4yz\:−\:5\:=\:0$ I asked my teacher, and he said I should use implicit ...
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1 vote
1 answer
84 views

find derivative of $\sqrt{x}+\sqrt{y}=\sqrt{a}$ [duplicate]

I have an Implicit Function $\sqrt{x}+\sqrt{y}=\sqrt{a}$ the graph of the function is I need to prove that $p+q=a$ and I need to find $\frac{d}{dx}$ to find the the slop to prove that. result: ...
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2 answers
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solving for $f(x)$:$\frac{d \ln f (x)}{d \ln x} = \frac{\alpha}{x}$

I have a continuously differentiable function $f(x)$ that satisfies $\frac{d \ln f (x)}{d \ln x} = \frac{\alpha}{x}$, with $f(0)=0$. How can I find $f(x)$?
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0 answers
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Expand the function by the Taylor formula in the vicinity of the point

Expand the function $f(x,y)=2x^2-xy-y^2-6x-3y+5$ by the Taylor formula in the vicinity of the point $(1,-2)$. I have an answer: But can you tell me the steps on how to do that? Am I right that we ...
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3 votes
3 answers
182 views

How to find du and dv?

Find $du$ and $dv$ if $u+v=x+y$ and $\frac{\sin(u)}{\sin(v)}=\frac{x}{y}$. How to solve this? Found almost an answer: But how do we get $du=...$ from the second?
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0 answers
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differentiation of explicit function vs implicit function

I have the following implicit function: $xy=5$. I want to get $y'$ or $\frac{dy}{dx}$. In this case, implicit differentiation gives me $\frac{dy}{dx}=\frac{-y}{x}$. I am curious why explicit ...
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  • 111
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1 answer
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Calculus: Triple derivative. Where am I going wrong?

$$x^2+xy+y^3=1$$Find $y'''$ at $x = 1$ My solution: $$y\Big|_{x = 1} = 0 \\ 2x + y + xy' + 3y^2y' = 0\\ 2 + y' + y' + xy'' + y'(6yy') + 3y^2y'' = 0 \\ 2 + 2y' + xy'' + 2y'6y + 3y^2y'' = 0 \\ y''\Big|...
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  • 25
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3 answers
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Derivative of $ \sqrt y + \sqrt x = 4 $ at $ ( 0.25 , 0.25 ) $

Find the derivative of $ \sqrt y + \sqrt x = 4 $ at $ ( 0.25 , 0.25 ) $. Finding derivative, I get $ \frac { \mathrm d y } { \mathrm d x } = - \sqrt { \frac y x } $. At $ ( 0.25 , 0.25 ) $, the value ...
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1 vote
0 answers
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Determine the location of $(x_0, y_0)$ in terms of $h$ and $L$ using Calculus ideas

This is part of a Calculus 1 project and I am sorely stuck on this part. Let's say you have a circle with center $(h,0)$ and radius $L$. This may or may not matter, but you can assume $h$ is positive ...
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