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2.3 First Derivatives of the Product of Two Polynomials

2.3 Find the Derivatives of a Product using Product Rule

(A) The Product Rule

Method 1

If u(x) and v(x) are two functions of x and y = uv then


Example:



Method 2 (Differentiate Directly)

Example:
Given that  y = ( 2 x + 3 ) ( 3 x 3 − 2 x 2 − x ) ,  find  d y d x

Solution:
y = ( 2 x + 3 ) ( 3 x 3 − 2 x 2 − x ) d y d x = ( 2 x + 3 ) ( 9 x 2 − 4 x − 1 ) + ( 3 x 3 − 2 x 2 − x ) ( 2 ) d y d x = ( 2 x + 3 ) ( 9 x 2 − 4 x − 1 ) + ( 6 x 3 − 4 x 2 − 2 x )

Practice 1:
Given that y = 4x3 (3x + 1)5, find dy/dx. 

Solution:
y = 4x3 (3x + 1)5
dy/dx 
= 4x3. 5(3x + 1)4.3 + (3x + 1)5.12x2
= 60x3 (3x + 1)4 + 12x2 (3x + 1)5
= 12x2(3x + 1)4 [5x  + (3x+ 1)]
= 12x2(3x + 1)4 (8x  + 1)

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