1. MTH1170 Implicit Differentiation

2. 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.

3. Example
Find y’
𝑦 2 = 𝑥 2

4. 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’.

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

6. Example
Find y’ for the following equation:

7. Example
Find y’ for the following equation:

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