Definition
Implicit Differentiation
Implicit differentiation is a method for computing the derivative
when the relationship between the variables and is given implicitly by an equation of the form: rather than explicitly as
. The method treats as an (unknown) function of , differentiates every term of with respect to while applying the chain rule to any term that involves , and then solves the resulting equation for .
Example
Let’s find
- Differentiate both sides with respect to
:
- Apply the power rule and chain rule to each term:
- Collect terms involving
on one side and the remaining terms on the other side:
- Factor out
from the left side:
- Solve for
:
- Simplify the expression by dividing numerator and denominator by 3:
Therefore, the derivative of