Product and Quotient Rules and Higher Order Derivatives

Product and Quotient Rules and Higher – Order Derivatives Section 2. 3

The Product Rule The derivative of fg is the first function times the derivative of the second, plus the second function times the derivative of the first.

Example: h(x) = (3 x – 2 x 4)(6 – 7 x) Find h’(x)

Example: d/dx [x cos x] =

Example: Find the derivative of y = 2 x sin x – 2 cos x

The Quotient Rule The derivative of f/g of two differentiable function f and g is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

Example:

Example: Find y’

Differentiate each function: f(x) = g(x) =

Derivatives of Trig Functions: Find the derivative of y = tan x Find the derivative of y = cot x

Derivatives of Trig Functions Find the derivative of y = sec x Find the derivative of y = csc x

Example: Differentiate each Trig function h(x) = x + cot x h(t) = (sec t)/t f(x) = sin x cos x

Higher – Order Derivatives: A velocity function is the of. An function is the derivative of. Thus, the function is a of the function.

Example: Finding acceleration due to gravity on the moon. Because the moon has no atmosphere, a falling object encounters no air resistance. The position function of each object on the moon is given by s(t) = -0. 81 t 2 + 2. Find the acceleration due to gravity on the moon.