IMPLICIT DIFFERENTIATION
(utilizing the Chain Rule)
explicit function:
a function in which the dependent variable is directly stated as a
function of the independent variable ex. 𝑦 = 𝑥 2 or 𝑦 = 𝑥
implicit function:
a function in which the dependent variable is NOT directly stated
as a function of the independent variable ex. 2𝑥𝑦 − 𝑦 3 = 4
Example 
1
Consider the function, 𝑦 = . Determine y’.
𝑥
(Explicit Version)
(Implicit Version)
IMPLICIT DIFFERENTIATION
If an equation defines y implicitly as a differentiable function of x,
the derivative of y with respect to x can be determined as follows:
1. Differentiate both sides of the equation wrt x.
2. Use the Chain Rule when differentiating y ex.
3. Solve for y’
Example 
𝑑𝑦
𝑑𝑥
Determine
𝑑𝑦
𝑑𝑥
.
for 𝑥 2 + 𝑦 2 = 1.
𝑑
𝑑𝑥
𝑦2 =
Example 
a)
Example 
Differentiate each of the following with respect to x:
2𝑥𝑦 − 𝑦 3 = 4
b)
𝑥 3 + 𝑦 3 = 6𝑥𝑦
Determine the equation of the tangent to 𝑦 5 + 𝑥 2 𝑦 − 2𝑥 2 = −1 at the
point −1,1 .
Homework:
p.564 #2, 3ac, 4, 6–9