Derivative Rules – Product Rule

Definition and Formula

Essentially, the product rule of calculus is as follows:

The derivative of f(x) * g(x) = f'(x) * g(x) + f(x) * g'(x)

f(x) and g(x) are the original functions being multiplied. f'(x) and g'(x) are the derivatives of f (x) and g(x), respectively.

The product rule is used when a function is composed of two functions being multiplied by each other.

Before doing some differentiation examples, one must determine f(x) and g(x).

Example 1 – What are f(x) and g(x) of h(x) = (6×3-12x)(3x+4)?

f(x) = (6x3 – 12x)
g(x) = (3x + 4)

Example 2 – What are f(x) and g(x) of h(x) = 6x23x4?

f(x) = x2
g(x) = 3x4

Example 3 – What are f(x) and g(x) of h(x) = x-2(4 + 4x-3)?

f(x) = x-2
g(x) = (4+4x-3)

Now that f(x) and g(x) are found, differentiation is possible.

Example 4 – What is the derivative of h(x) = (6x3-12x)(3x+4)?

f(x) and g(x) have already been determined:

f(x) = (6x3-12x)
g(x) = (3x+4)

f'(x) and g'(x) must be determined. Applying the power rule to f(x) and g(x),

f'(x) = (3)*6x3-1-(1)*12x1-1 = 18x2-12x0 = 18x2 = 12
g'(x)= (1)*3x1-1 + 0 = 3x0 = 3

According to the product rule,

h’(x) = (6x3 – 12x)* 3 + 12 * (3x + 4)

Simplified,

h’(x) = 18x3 – 36x + 36x + 48 = 18x3 + 48

Example 5 – What is the derivative of h(x) = 6x23x4?

f(x) and g(x) have already been determined:

f(x) = x2
g(x) = 3x4

f'(x) and g'(x) must be determined. Applying the power rule to f(x) and g(x),

f’(x) = (2)*x2-1 = 2x1 = 2x
g’(x)= (4)*3x4-1 = 12x3

According to the product rule,

h’(x) = x2*12x3 + 3x4 * 2x

Simplified,

h’(x) = 12x3+2 + 6x4+1 = 12x5 + 6x5

Example 6 – What is the derivative of h'(x) = x-2(4 + 4x-3)?

f(x) and g(x) have already been determined:

f(x) = x-2
g(x) = (4 + 4x-3)

f’(x) and g’(x) must be determined. Applying the power rule to f(x) and g(x),

f’(x) = (-2)*x-2-1 = -2x-3 = -2 1/x3
g’(x)= 0 + (-3) * 4x-3-1 = -12x-4 = -12 1/x4

According to the product rule,

h’(x) = x-2(-12 1/x4) + (4 + 4x-3)(-2 1/x3)

Simplified,

h’(x) = -12x-2-4 + -2*4*1/x3+ -2*-4x-3-3 = -12x-6-8x3 + 8x-6 = -12x-6 – 8 1/x3 + 8 1/x6