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 If n is any real number and x ≠ 0, then
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: 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
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
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, ), (0.948, 6.508), (3.539, )

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. Higher-Order Derivatives
f’ is called the first derivative of f
f(n) is called the nth 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, odd, 25, 27, 29, odd, 46)