HIGHER ORDER DERIVATIVES Product & Quotient Rule

The Product Rule Theorem. Let f and g be differentiable functions. Then the derivative of the product fg is (fg) '(x) = f(x) g '(x) + g(x) f '(x) In other words, first times the derivative of the second plus second times the derivative of the first.

Using the Product Rule Example:

Product Rule Example:

Product Rule Example

Quotient Rule Theorem. Let f and g be differentiable functions. Then the derivative of the quotient f/g is In other words, low d high minus high d low over low low.

Quotient Rule Example Find the derivative of

Rewriting Before Differentiating Example:

Derivatives of Trigonometric Functions

Proof of Derivative of Tangent Considering

Differentiating Trig Functions y = x – tan x y = x sec xy’ = x(sec x tan x) + sec x (1) = sec x (x tan x + 1)

Differentiating Trigonometric Functions

Higher Order Derivatives Applications: Finding Acceleration Due to Gravity Population Growth p. 125 problem 79 Any time we are asked to find the rate at which a rate is changing, this is a second derivative.