# Quotient Rule in Differentiation

How to solve the quotient rule in differentiation problems: formula, proof, example, and its solution.

## Formula

[*f*(*x*)/*g*(*x*)]' = [*f*'(*x*)*g*(*x*) - *f*(*x*)*g*'(*x*)] / [*g*(*x*)]^{2}

## Proof

Think *f*(*x*)/*g*(*x*) as *f*(*x*) ⋅ 1/*g*(*x*).

Then [*f*(*x*)/*g*(*x*)]' = [*f*(*x*) ⋅ 1/*g*(*x*)]'.

Use the product rule.

The differentiation of 1/*g*(*x*) is -*g*'(*x*)/[*g*(*x*)]^{2}.

Reciprocal rule in differentiation

To subtract these two fractions,

multiply *g*(*x*)/*g*(*x*) to *f*'(*x*)/*g*(*x*).

Adding and subtracting rational expressions

Then [*f*(*x*)/*g*(*x*)]' = [*f*'(*x*)*g*(*x*) - *f*(*x*)*g*'(*x*)] / [*g*(*x*)]^{2}.

## Example

*y* = 4*x*/(*x*^{2} - 3*x*)*y*' is equal to,

the derivative of 4*x*, (4⋅1*x*^{0})

times (*x*^{2} - 3*x*)

minus (4*x*),

times, the derivative of *x*^{2} - 3*x*, (2*x*^{1} - 3⋅1*x*^{0})

over (*x*^{2} - 3*x*)^{2}.

Derivatives of polynomials