MTH1170 Implicit Differentiation

Preliminary We know how to find the derivatives of functions that pass the vertical line test, but what about graphs with more than one y value for a given x? We need to implicitly differentiate them by implying that y is a function of x, and applying the chain rule.

Example Find y’ 𝑦 2 = 𝑥 2

How to Implicitly Differentiate Consider y as one or more function of x. Differentiate both sides of the equation with respect to x. Because y is being considered a function of x, we need to use the chain rule to find it’s derivative. Everywhere you see a y in the equation, there will be a y’ produced by using the chain rule. Rearrange the equation to solve for y’.

Example Find the slope of a tangent line to the following function at the point P(3, -4).

Example Find y’ for the following equation:

Example Find y’ for the following equation:

Example Find the equation of a line tangent to the given equation at the provided point.