Implicit differentiation of a function of two or more variables

 Goal: find the derivative of an implicit function of two or more variables.

Let's consider the function defined by the ellipse 

x² + 3y² + 4y - 4 = 0

  Let's find its derivative. This is the implicit function of an ellipse defined by the following graph:

 

 Let's find the derivative of this function by taking the derivative of both sides:

 

This last expression of the derivative is the simplification of dy/dx = -2x/6y + 4

We can notice that the numerator is the partial derivative of the function f(x,y) = 0 with respect to x. The denominator is the partial derivative of f with respect to y. This fact leads to the following theorem:

Theorem 

Suppose that the function z = f(x,y) defines y implicitly as a function y = g(x) via the equation f(x,y) = 0, then

 provided 

 

If the equation f(x, y, z) = 0 defines z implicitly as a function  differentiable of x and y, then

 

Example

a. Calculate dy/dx if y is expressed implicitly as a function of x via the equation 3x² - 2xy + y² + 4x-6y - 11 = 0. What is the equation of the tangent line to the graph of this curve at point (2, 1)?

b' Calculate ẟz/ẟx and ẟz/ẟy given

Solution

Let's write f(x, y) = 3x² - 2xy + y² + 4x-6y - 11 = 0 and calculate ẟf/ẟx and ẟf/ẟy

ẟf/ẟx = 6x - 2y + 4  ẟf/ẟy = -2x + 2y - 6

The derivative is given by:

The slope of the tangent line at the point (2, 1) is given by:

The equation of the tangent line is given by:

This is the graph of the rotated ellipse represented by the equation 3x² - 2xy + y² + 4x-6y - 11 = 0

b. We have  

Let's calculate the partial derivatives of f with respect to x, y and z.

Finally let's calculate  ẟz/ẟx and ẟz/ẟy:

Practice

Find the derivative dy/dx of the function y defined implicitly as a function of x and y by the function 

What is the equation of the tangent line to the graph of this curve at point (3,2)?