1. Section 3.5 Implicit Differentiation 1

2. Example If f(x) = (x 7 + 3x 5 – 2x 2 ) 10, determine f ’(x). Now write the answer above only in terms of y if y = x 7 + 3x 5 – 2x 2. Answer: f ΄ (x) = 10(x 7 + 3x 5 – 2x 2 ) 9  (7x 6 + 15x 4 – 4x) Answer: f ΄ (x) = 10y 9  y΄

3. Examples If y is some unknown function of x, find

4. Purpose 9x + x 2 – 2y = 5 5x – 3xy + y 2 = 2y Easy to solve for y and differentiate Not easy to solve for y and differentiate Process wise, simply take the derivative of each side of the equation with respect to x and when we encounter terms containing y, we use the chain rule. In equations like 5x – 3xy + y 2 = 2y, we simply assume that y = f(x), or some function of x which is not easy to find.

5. Example y 3 = 2x Solving for y’, we have the derivative

6. Example x 2 y 3 = -7 Solving for y’, we have

7. Differentiate both sides of the equation. Since y is a function of x, every time we differentiate a term containing y, we need to multiply it by y’ or dy/dx. Solve for y’. Every term containing y’ should be moved to the left by adding or subtracting terms only. Every term containing no y’ should be moved to the right hand side. Factor out y’ and divide both sides by the expression inside ( ). Implicit Differentiation

8. Examples Determine dy/dx for the following. Find the equation of tangent line to the curve.

9. Example Find the derivative for

10. Derivative of Trig functions

11. Examples Find the derivative for each function.

12. Examples Find and simplify dy/dx for each function.