# 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 ...
1 vote
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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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### 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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### 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 ...
1 vote
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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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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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### 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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### 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 vote
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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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### 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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### 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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### 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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### 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 ... ### 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 ...