# How to Perform Implicit Differentiation on $$x^2-4xy+y^2=4$$?

• MHB
• karush
In summary, you found the derivative of y with respect to x by taking the implicit derivative of the given equation and solving for y'. You then factored out the common factor of -4x+2y and isolated y'. The final result is y' = (-x+2y)/(-2x+y).
karush
Gold Member
MHB
$\tiny{166.2.6.5}$
Find y'
$$x^2-4xy+y^2=4$$
dy/dx
$$2x-4(y+xy')+2yy'=2x-4y-4xy'+2yy'=0$$
factor
$$y'(-4x+2y)=-2x+4y=$$
isolate
$$y'=\dfrac{-2x+4y}{-4x+2y} =\dfrac{-x+2y}{-2x+y}$$

typo maybe not sure if sure if factoring out 4 helped

Last edited:
karush said:
$\tiny{166.2.6.5}$
Find y'
$$x^2-4xy+y^2=4$$
dy/dx
$$2x-4(y+xy')+2yy'=2x-4y-4xy'+2yy'=0$$
factor
$$y'(-4x+2y)=-2x+4y=$$
isolate
y'=\dfrac{-2x+4y}{-4x+2y}
=\dfrac{-x+2y}{-2x+y}\$

typo maybe not sure if sure if factoring out 4 helped

What you've done is correct.

## What is implicit differentiation?

Implicit differentiation is a method used to find the derivative of a function when it is not possible to express the dependent variable explicitly in terms of the independent variable.

## When is implicit differentiation used?

Implicit differentiation is used when the given function is not in the form of y = f(x) and cannot be easily differentiated using the power rule or other basic differentiation rules.

## What is the process of implicit differentiation?

The process of implicit differentiation involves taking the derivative of both sides of an equation with respect to the independent variable, treating the dependent variable as a function of the independent variable.

## What is the result of implicit differentiation?

The result of implicit differentiation is an expression for the derivative of the dependent variable in terms of the independent variable and the derivative of the dependent variable with respect to the independent variable.

## What are some common applications of implicit differentiation?

Implicit differentiation is commonly used in physics and engineering to find rates of change, tangents and normals to curves, and optimization problems. It is also used in economics and finance to analyze supply and demand functions and cost functions.

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