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) = (3x – 2x4)(6 – 7x)

Find h’(x)

Example:

d/dx [x cos x] =

Example:

Find the derivative of y = 2x 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:

2

233x

x

dx

d

Example: Find y’

4

/12

x

xy

Differentiate each function:

f(x) =

g(x) =

5

22 xx

x

xx

4

)42(2 2

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 functionh(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) = -0.81t2 + 2. Find the acceleration due to gravity on the moon.