3.3 Rules for Differentiation

1. 3.3 Rules for Differentiation AKA “Shortcuts”

2. Review from 3.2 • 4 places derivatives do not exist: • Corner • Cusp • Vertical tangent (where derivative is undefined) • Discontinuity (jump, hole, vertical asymptote, infinite oscillation) • In other words, a function is differentiable everywhere in its domain if its graph is smooth and continuous.

3. 3.2 Intermediate Value Theorem for Derivatives • If a and b are any two points in an interval on which f is differentiable, then f’ takes on every value between f’(a) and f’(b).

4. Derivatives of Constants • Find the derivative of f(x)= 5. Derivative of a Constant: If f is the function with the constant value c, then, (the derivative of any constant is 0)

5. Power Rule • What is the derivative of f(x) = x3? • From class the other day, we know f’(x) = 3x2. • If n is any real number and x ≠ 0, then In other words, to take the derivative of a term with a power, move the power down front and subtract 1 from the exponent.

6. Power Rule • Example: • What is the derivative of • Example: • What is the derivative of

7. Power Rule • Example: • What is the derivative of Now, use power rule

8. Constant Multiple Rule • Find the derivative of f(x) = 3x2. Constant Multiple Rule: If u is a differentiable function of x and c is a constant, then 0 In other words, take the derivative of the function and multiply it by the constant.

9. Sum/Difference Rule • Find the derivative of f(x) = 3x2 + x Sum/Difference Rule: If u and v are differentiable functions of x, then their sum and difference are differentiable at every point where u and v are differentiable. At such points, In other words, if functions are separated by + or –, take the derivative of each term one at a time.

10. Example • Find where horizontal tangent occurs for the function f(x) = 3x3 + 4x2 – 1. A horizontal tangent occurs when the slope (derivative) equals 0.

11. Example • At what points do the horizontal tangents of f(x)=0.2x4 – 0.7x3 – 2x2 + 5x + 4 occur? Horizontal tangents occur when f’(x) = 0 To find when this polynomial = 0, graph it and find the roots. Substituting these x-values back into the original equation gives us the points (-1.862, -5.321), (0.948, 6.508), (3.539, -3.008)

12. Product Rule • If u and v are two differentiable functions, then Also written as: In other words, the derivative of a product of two functions is “1st times the derivative of the 2nd plus the 2nd times the derivative of the 1st.”

13. Product Rule • Example: Find the derivative of

14. Quotient Rule • If u and v are two differentiable functions and v ≠ 0, then Also written as: In other words, the derivative of a quotient of two functions is “low d-high minus high d-low all over low low.”

15. Quotient Rule • Example: Find the derivative of

16. f(n)is called the nth derivative of f Higher-Order Derivatives • f’ is called the first derivative of f • f'' is called the second derivative of f • f''' is called the third derivative of f

17. Higher-Order Derivatives • Example Find the first four derivatives of

18. Friday Classwork: • Section 3.3 • (#1-9 odd, 15-23 odd, 25, 27, 29, 33-37 odd, 46)