1. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Do Now: Aim: To memorize more stuff about differentiation: Product/quotient rules and more!!!!

2. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Two Helpful Basics

3. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus The Product Rule The derivative of the product of two differentiable functions f and g is itself differentiable. The derivative of fg is the first function times the derivative of the second plus the second function times the derivative of the first. Find the derivative of h(x) = (3x – 2x 2 )(5 + 4x) = -24x 2 + 4x + 15 first derivative of second second derivative of first

4. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus The Quotient Rule The derivative of the quotient of two differentiable functions f and g is itself differentiable at all values of x for which g(x)  0. The derivative of f/g is given by the denominator times the derivative of the numerator minus the numerator time the derivative of the denominator, all divided by the square of the denominator.

5. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus The Quotient Rule Find the derivative of denom. square of denom. derivative of numer. numer.derivative of denom.

6. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Model Problems Find the derivative of simplify rewrite to eliminate complex nature

7. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Model Problems Rewriting Quotients to utilize the Constant Multiply Rule reduces work required. rewrite to eliminate complex nature differentiatesimplify

8. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Model Problems

9. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Do Now: Aim: To memorize more stuff about differentiation: Product/quotient rules and more!!!! Find the derivative

10. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Derivatives of Trig Functions Differentiate both sides individually left side

11. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Derivatives of Trig Functions Differentiate both sides individually Show two derivatives are equal right side

12. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Higher Order Derivatives First Derivative: y’, f’(x), Second Derivative: y’’, f’’(x), Third Derivative: y’’’, f’’’(x), n th Derivative: y (n), f (n) (x), s’(t) = v(t) Velocity Function s’’(t) = v’(t) = a(t) Acceleration Function Position Function

13. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Model Problem Because the moon has no atmosphere, a falling object on the moon encounters no air resistance. In 1971, astronaut David Scott demonstrated that a feather and a hammer fall at the same rate on the moon. The position function for each of these falling objects is given by s(t) = -0.81t 2 + 2 where s(t) is the height in meters and t is the time in seconds. What is the ratio of the earth’s gravitational force to the moon’s?

14. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Model Problem s(t) = -0.81t 2 + 2 Position Function s’(t) = v(t) = -1.62t Velocity Function s’’(t) = v’(t) = a(t) = -1.62 Acceleration Function Acceleration due to gravity on the moon is -1.62 meters per second per second. Acceleration due to gravity on earth is -9.8 meters per second per second.

15. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus nDeriv( A ball is thrown straight up into the air, and the height of the ball above the ground is given by the function h(t) = 6 + 37t – 16t 2, where h is in feet and t is in seconds. What is the velocity of the ball at time t = 3.2? MATH 8 x, ENTER 6 + 37t – 16t 2, 3.2 ENTER

16. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Model Problem Find the derivative Find the derivative when x = 2 using nDeriv( function of calculator

17. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Model Problem Find the derivative Find the derivative when x = 3 using nDeriv( function of calculator

18. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Model Problem Find the derivative Find the derivative when x = 3 using nDeriv( function of calculator

19. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Model Problem Find an equation of the tangent line to the graph Use nDeriv( function of calculator

20. Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Model Problem Find the derivative