1. 4.2 – Implicit Differentiation
Ch. 4 – More Derivatives
4.2 – Implicit Differentiation

2. Ex: Find the derivative of x2 + y2 = 25.
This function isn’t solved for y, so we will use IMPLICIT DIFFERENTIATION!
Implicit Differentiation: Differentiating when the dependent variable is not solved for. Follow these steps:
Take the derivatives term by term. Use normal derivative rules for all variables and numbers.
x2 and y2 are variables, but 25 is a constant…
When you take the derivative of a y, include a dy/dx (or a y’) afterward.
Factor out and solve for dy/dx!

3. Ex: Find the derivative of 4x2 + 2xy = y2.
Take the derivatives term by term. Use normal derivative rules for all variables and numbers.
Don’t forget about product rule for the 2xy term!
When you take the derivative of a y, include a dy/dx (or a y’) afterward.
Factor out and solve for dy/dx!
Get all the dy/dx’s to one side and factor!

4. Ex: Find the slope of the line tangent to 4x2 y – 6y = x3 + 2 at (2, 1).
I’ll use y’ instead of dy/dx for this problem just to show you that they are interchangeable:
Don’t forget about product rule for 4x2 y!
Now evaluate at the point (2, 1)…

5. Ex: Differentiate x = cosy.
Ex: Differentiate (2x + y2)3 – y = 8.
Use chain rule!

6. Ex: Differentiate

7. Ex: Find the 2nd derivative of y2 = 4x .
Find the 1st derivative and solve for y’…
Now take the 2nd derivative (remember, we still include a dy/dx for derivatives of y)…
Lastly, substitute in your previously discovered dy/dx…

8. Ex: Find the 2nd derivative of x3 +y2 = 9 .
Find the 1st derivative and solve for y’…
Now take the 2nd derivative (remember, we still include a dy/dx for derivatives of y)…
Lastly, substitute in your previously discovered dy/dx…