1. 3.5 – Implicit Differentiation

2. Try It If f(x) = (x7 + 3x5 – 2x2)10, determine y΄.
Answer: f΄(x) =10(x7 + 3x5 – 2x2)9(7x6 + 15x4 – 4x)
Now write the answer above only in terms of y if y = (x7 + 3x5 – 2x2).
Answer: f ΄(x) = d[y10]/dx = 10y9y΄

3. Try It If y is some unknown function of x, determine d[y35] / dx.
Answer: d[y35] / dx =_____________
In general, if y is some function of x, then what is d[yn] / dx?
Answer: d[yn] / dx = ______________

4. Power Rule for Implicit Differentiation
If y is some unknown function of x (or difficult to determine), then
using the chain rule.

5. Purpose Which of these would be easy to solve for y and differentiate?
9x + x2 – 2y = 5
5x – 3xy + y2 = 2y
Easy to solve for y and differentiate
Not easy to solve for y and differentiate

6. The Idea
In equations like 5x – 3xy + y2 = 2y, we simply assume that y = f(x), or some function of x which is not easy to find.
Process wise, simply take the derivative of each side of the equation with respect to x and when we encounter a yn, we use

7. Examples Determine dy/dx for the following. 1. y3 = 2x 2. x2y3 = –7
3. y1/2 – 3x1/3y1/4 = 2x
With products of yn and xn, you must use the product rule. Why? This is also true for quotients, but with the quotient rule.

8. You Try It
Determine dy/dx for the following.

9. Derivative of Inverse Trig Functions

10. Derivative of Inverse Trig Functions

11. Derivative of Inverse Trig Functions