# Derivative of a Quotient

How to find the derivative of a quotient f(x)/g(x): formula (quotient rule), 1 example, and its solution.

## FormulaQuotient Rule

The derivative of f(x)/g(x) is -[f'(x)g(x) - f(x)g'(x)]/[g(x)]^{2}.

## Exampley = 4x/(x^{2} - 3), y' = ?

4x/(x^{2} - 3) is the quotient of 4x and (x^{2} - 3).

So y' is equal to ...

Write - and the fraction bar.

In the numerator,

write, the derivative of 4x, 4.

Derivative of a Polynomial

Write the denominator (x^{2} - 3).

Write -.

Write the numerator 4x.

Write, the derivative of (x^{2} - 3), (2x^{1} - 0).

And in the denominator,

write, the square of the denominator (x^{2} - 3), (x^{2} - 3)^{2}.

So y' = -[4⋅(x^{2} - 3) - 4x(2x^{1} - 0)]/(x^{2} - 3)^{2}.

4⋅(x^{2} - 3) = 4x^{2} - 12

-4x(2x^{1} - 0) = -8x^{2}

4x^{2} - 8x^{2} = -4x^{2}

-(-4x^{2} - 12) = 4x^{2} + 12

So

y' = [4x^{2} + 12]/(x^{2} - 3)^{2}

is the derivative of the given function.