## 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) = (6x^{3} – 12x)

g(x) = (3x + 4)

**Example 2 – What are f(x) and g(x) of h(x) = 6x ^{2}3x^{4}?**

f(x) = x^{2}

g(x) = 3x^{4}

**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) = (6x ^{3}-12x)(3x+4)?**

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

f(x) = (6x^{3}-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)*6x^{3-1}-(1)*12x^{1-1} = 18x^{2}-12x^{0} = 18x^{2} = 12

g'(x)= (1)*3x^{1-1} + 0 = 3x^{0} = 3

According to the product rule,

h’(x) = (6x^{3} – 12x)* 3 + 12 * (3x + 4)

Simplified,

h’(x) = 18x^{3} – 36x + 36x + 48 = 18x^{3} + 48

**Example 5 – What is the derivative of h(x) = 6x ^{2}3x^{4}?**

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

f(x) = x^{2}

g(x) = 3x^{4}

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^{1} = 2x

g’(x)= (4)*3x^{4-1} = 12x^{3}

According to the product rule,

h’(x) = x^{2}*12x^{3} + 3x^{4} * 2x

Simplified,

h’(x) = 12x^{3+2} + 6x^{4+1} = 12x^{5} + 6x^{5}

**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/x^{3}

g’(x)= 0 + (-3) * 4x^{-3-1} = -12x^{-4} = -12 1/x^{4}

According to the product rule,

h’(x) = x^{-2}(-12 1/x^{4}) + (4 + 4x^{-3})(-2 1/x^{3})

Simplified,

h’(x) = -12x^{-2-4} + -2*4*1/x^{3}+ -2*-4x^{-3-3} = -12x^{-6}-8x^{3} + 8x^{-6} = -12x^{-6} – 8 1/x^{3} + 8 1/x^{6}