DO NOW: Write each expression as a sum of powers of x:

3.3 - Rules for Differentiation HW: Pg. 124 #2-22e

Derivative of a Constant Function If f is the function with the constant value c, then EX: f(x) = 4 f’(x) = 0

Power Rule for Positive Integer Powers of x If n is a positive integer, then EX: f(x) = x3 + x2

The Constant Multiple Rule If f(x) is a differentiable function of x and c is a constant, then EX: f(x) = 5x5

The Sum and Difference Rule If f(x) and g(x) are differentiable functions of x, then their sum and difference are differentiable at every point where f(x) and g(x) are differentiable. At such points,

Example 1 Find dp/dt if p(t) = t4 + 5t3 - 2t2 + 6/7t + 4

Example 2 Does the curve y = x4 - 2x2 + 2 have any horizontal tangents?

Example 3 Differentiate: (a) (b)

Example 4 Find the derivative of f(x) = (x + 4)(x - 5)

The Product Rule The product of two differentiable functions f(x) and g(x) is differentiable, and “The derivative of two functions equals the first function times the derivative of the second plus the second function times the derivative of the first”

Example 4 (cont) Find the derivative of f(x) using the product rule: F(x) = (x + 4)(x - 5)

Example 5 Find f’(x) if f(x) = (x3 + 4)(x2 + 5)

The Quotient Rule At a point where g(x) ≠ 0, the quotient y = f(x)/g(x) of two differentiable functions is differentiable, and

Example 6 Find f’(x) if

Example 7 Find the equation of the line tangent to the curve y = (x2 + 3)/2x at the point (2,1).

Second and Higher Order Derivatives

Example 8 Find the first four derivatives of y = x8 + 12x5 - 4x4 + 10x3 - 6x + 5

Example 9 If f(x) = (√x)g(x) , g(4) = 2 and g’(4) = 3

Example 10 Let y = uv be the product of the functions u and v. Find y’(2) if u(2) = 3, u’(2) = -4, v(2) = 1, and v’(2) = 2.