2.3 The Product Rule
Unit 2: Derivatives
MCV 4U1: Calculus and Vectors
Investigate
Find the derivative of the following function listed in two equivalent forms:
( )
( )
(
)(
)
Product Rule
If f(x) and g(x) are differentiable at x and p(x) = f(x)g(x) then:
( )
( ) ( )
[ ( ) ( )]
(
( )
( ) ( )
( )
( )
( )
)
Example: Differentiate using the product rule for the function ( )
(
Example: Find the equation of the tangent line to
Example: Consider the function ( )
a) Differentiate and simplify
√ (
√ )(
(
)(
).
) at the point P (1, 4).
)
b) Find the values of x where the function is not differentiable. Explain.
c) Find the values of x where the tangent line is horizontal.
Product of Three Functions
If f(x), g(x) and h(x) are differentiable at x and p(x) = f(x)g(x)h(x) then:
( )
( ) ( ) ( )
( ) ( ) ( )
Example: Differentiate but do not simplify ( )
Power of a Function Rule (Chain Rule)
. If ( )
If f(x) is differentiable at x, then so is
( )
(
[ ( )]
)(
[ ( )] then,
( )
)
(
Example: Differentiate
Example: Given ( )
(√
( ) ( ) ( )
(
Page 90-93 #1, 2, 5alt, 6, 7, 8a, 13
)
) , find the velocity at t = 1s.
)(
)