## Definition and Formula

Essentially, the quotient rule of calculus is as follows:

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) = ?**

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)*x^{1-1}-0 = x^{0} = 1

g’(x) = (1)*x^{1-1}+0 = x^{0} = 1

According to the quotient rule,

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

f(x) = ln(x)

g(x) = 2x^{2}

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

f’(x) = 1/x (the derivative of ln(u) = 1/u)

g’(x) = (2)*2x^{2-1} = 4x^{1} = 4x

According to the quotient rule,

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

f(x) = 1-x^{2}

g(x) = 5x^{2}

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

f’(x) = 0-(2)*x^{2-1} = -2x^{1} = -2x

g’(x) = (2)*5x^{2-1} = 10x^{1} = 10x

According to the quotient rule,

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

Applying the power rule,

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