2. 3.3 Rules for Differentiation Colorado National Monument

3. If the derivative of a function is its slope, then for a constant function, the derivative must be zero. example: The derivative of a constant is zero. Consider the function y = 4 graphed to the right. Clearly, the slope of all the tangent lines on this graph would all be 0.

4. If we find derivatives with the difference quotient: We observe a pattern: …

5. examples: power rule We observe a pattern:…

6. Differentiate

7. examples: constant multiple rule: When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.

8. (Each term is treated separately) constant multiple rule: sum and difference rules:

10. Example: Find the x values of the horizontal tangents of: Horizontal tangents occur when slope = zero.

15. First derivative (slope) is zero at:

16. product rule: Notice that this is not just the product of two derivatives.

17. Since multiplication is commutative, the product rule order can change. The book uses this one. I do not. another way to remembered it : f(x) g(x) so der. of 1 st times the 2 nd + 1 st times the der. of the 2 nd

18. quotient rule: or Since it is subtraction, the order does matter

19. Same as product, but subtraction in the numerator

20. Find the equation of the tangent line to the curve

21. What is the equation of the normal line to f(x) = at x = –4?

22. What is the velocity as a function of time (in seconds) of the time-position function meters?

23. Example: working with numerical values Let y = uv be the product of the functions u and v. Find y’(2) if Ans: Find the product y’ = u’ v + u v’ y’(2)= (-4)(1)+(3)(2) = -4+6 = 2

24. Higher Order Derivatives: is the first derivative of y with respect to x. is the second derivative. (y double prime) is the third derivative.is the fourth derivative. We will learn later what these higher order derivatives are used for. 

25. Ex.

26. the end