1. 4-3 Derivative of a Quotient of Two Functions

2. This chapter allows you to solve “catching up problems” in which one moving object wants to reach another moving object.This chapter allows you to solve “catching up problems” in which one moving object wants to reach another moving object.

3. Property: Derivative of a Quotient of Two Functions: – –If y = u/v, where u and v are differentiable functions, and v doesn’t equal 0, then u’v – uv’ y’ = v^2 Words – Derivative of top times bottom, minus top times derivative of bottom, all divided by bottom squared

4. In order to differentiate, or find the derivative, some functions, its important to remember the Property: Derivatives of Sine and Cosine Functions.In order to differentiate, or find the derivative, some functions, its important to remember the Property: Derivatives of Sine and Cosine Functions. –d/dx(sinx) = cosX –d/dx(cosx) = -sinX The chain rule, explained in 3-7, is applied in numerous problems when finding the derivative.The chain rule, explained in 3-7, is applied in numerous problems when finding the derivative.

5. Differentiate:Differentiate: (5x – 2)^7 (5x – 2)^7 y = (4x +9)^3 Differentiate:Differentiate: d ( 5 ) d ( 5 ) dx ( 7x^3) dx ( 7x^3) Differentiate:Differentiate: sin5x sin5x f(x) = 8x - 3 f(x) = 8x - 3

6. Detailed Procedure Problem # 1:Problem # 1: (5x – 2) 7 (5x – 2) 7 y= (4x +9) 3 y= (4x +9) 3 Use the Property to solve problem (HINT: Derivative of top times bottom, minus top times derivative of bottom, all divided by bottom squared.) The chain rule, as in 4-2, also is applied when finding derivatives y’ = 7(5x-2)^6(5) * (4x+9)^3 - (5x - 2)^7 * 3(4x + 9)^2(4) (4x + 9)^6 = (5x - 2)^6 (80x + 339) (4x + 9)^6 (4x + 9)^6

7. Differentiate:Differentiate: sin5x sin5x f(x) = 8x - 3 f(x) = 8x - 3 1.) cos5x(5)(8x - 3) - sin5x(8) 1.) cos5x(5)(8x - 3) - sin5x(8) f’x = (8x - 3)^2 f’x = (8x - 3)^2 2.) 5cos5x(8x - 3) - 8sin5x 2.) 5cos5x(8x - 3) - 8sin5x f’x = (8x - 3)^2 f’x = (8x - 3)^2

8. Black Hole Problem :Black Hole Problem : Ann Astronaut’s spaceship gets trapped in the gravitational field of a black hole! Her velocity, v(t) miles per hour is given: Ann Astronaut’s spaceship gets trapped in the gravitational field of a black hole! Her velocity, v(t) miles per hour is given: 1000 1000 v(t) = 3 - t v(t) = 3 - t a.) How fast is she going when t=1? When t = 2? When t = 3? a.) How fast is she going when t=1? When t = 2? When t = 3? B.) Write an equation for the acceleration function, a(t). B.) Write an equation for the acceleration function, a(t). C.) What is her acceleration when t=1? When t = 2? When t = 3? C.) What is her acceleration when t=1? When t = 2? When t = 3?

9. A.) a(1) = 250 mph/hrA.) a(1) = 250 mph/hr a(2) = 1000 mph/hr a(2) = 1000 mph/hr a(3) = 1000/0 mph/hr = no answer a(3) = 1000/0 mph/hr = no answer B.) 1000B.) 1000 a(t) = (3 - t)^2 a(t) = (3 - t)^2 C.) a(1) = 250 mph/hrC.) a(1) = 250 mph/hr a(2) = 1000 mph/hr a(2) = 1000 mph/hr a(3) = 1000/0 mph/hr = no answer a(3) = 1000/0 mph/hr = no answer

10. This section taught you how to find an equation for the derivative function in one step when the function contains a quotient of two other functions.This section taught you how to find an equation for the derivative function in one step when the function contains a quotient of two other functions. The Property; Derivative of a Quotient of Two Functions was discussed and applied.The Property; Derivative of a Quotient of Two Functions was discussed and applied.