1. 2.3/ The Product & Quotient Rules and Higher Order Derivatives
By Annie Kuo
Period 4th

2. In Section 2.2, you learned the basic differentiation of x^n by bringing the exponent down in front of x and subtract the exponent by 1.
d/dx[x^n]=nx^(n-1)
But, what if you have a problem like this??

3. This is why you have the PRODUCT RULE and the QUOTIENT RULE.

4. PRODUCT RULE
When you multiply two differential functions, the derivative of the PRODUCT equals the first function times the derivative of the second PLUS the second times the derivative of the first.

5. PROBLEMO #1: Differentiate

9. THE QUOTIENT RULE
When you divide two differentiable function, the QUOTIENT equals the bottom (denominator) function times by the derivative of the top (numerator) MINUS top times derivative of the bottom ALL OVER the bottom squared.

10. Low D’high minus High D’Low over Low Low
OR, you can sing a rhyme:
Low D’high minus High D’Low over Low Low

11. PROBLEMO #2: Differentiate

14. HIGHER ORDER DERIVATIVES
Last, but not the least, here are the symbolic representation of HIGHER ORDER DERIVATIVES.

15. What is the point??
Higher Order Derivatives can be applied in real life to determine position, velocity, and acceleration function.
s(t) Position
v(t)=s’(t) Velocity
a(t)=v’(t)=s’’(t) Acceleration

16. PROBLEMO #3: Real Life Application
At the Annual Calculus Car Race, Bob
wants to test out his car’s acceleration rate. Where s (t) represents the distance in meters and t represents the time in seconds, find the car’s acceleration with the given position function:
s(t)= 0.25t ^3+3

18. Now, let’s set aside the mathematics for one second…
Sir Isaac Newton
( )
“Father of Calculus”

19. PROBLEMO 4: Mix Review

22. PROBLEMO #5: Real Life Application
At the Annual Calculus Car Race, Billy Joe
wants to test out his car’s acceleration rate. Where s (t) represents the distance in meters and t represents the time in seconds, find the car’s acceleration with the given position function:
s(t)= 11t ^2+4t

24. Thank you for your attention in this presentation!
That’s all folks!
Thank you for your attention in this presentation!