Derivative Rules – Quotient Rules

Definition and Formula

Essentially, the quotient rule of calculus is as follows:

Quotient 1

f(x) and g(x) are the original functions being divided. df(x) and dg(x), or f’(x) and g’(x), are the derivatives of f(x) and g(x), respectively.

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

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

Example 1 – Find the derivative of h(x) = Quotient 2?

f(x) = x-1
g(x) = x+2

f(x) and g(x) must be differentiated. Applying the power rule to both functions,

f’(x) = (1)*x1-1-0 = x0 = 1
g’(x) = (1)*x1-1+0 = x0 = 1

According to the quotient rule,

Quotient 3

Example 2 – Find the derivative of h(x) = Quotient 4

f(x) = ln(x)
g(x) = 2x2

f(x) and g(x) must be differentiated.

f’(x) = 1/x (the derivative of ln(u) = 1/u)
g’(x) = (2)*2x2-1 = 4x1 = 4x

According to the quotient rule,

Quotient 5

Example 3 – Find the derivative of h(x) = Quotient 6

f(x) = 1-x2
g(x) = 5x2

f(x) and g(x) must be differentiated. Applying the power rule to f(x) and g(x)

f’(x) = 0-(2)*x2-1 = -2x1 = -2x
g’(x) = (2)*5x2-1 = 10x1 = 10x

According to the quotient rule,

Quotient 7

As an alternative, you can simplify h(x) first into a single polynomial:

Quotient 8

Applying the power rule,

Quotient 9

Leave A Comment