Derivative of a Quotient

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

Formula

Derivative of f(X)/g(x): Quotient Rule

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

Example

Example

Solution

4x/(x2 - 3) is the quotient of 4x and (x2 - 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 (x2 - 3).

Write -.

Write the numerator 4x.

Write, the derivative of (x2 - 3), (2x1 - 0).

And in the denominator,
write, the square of the denominator (x2 - 3), (x2 - 3)2.

So y' = -[4⋅(x2 - 3) - 4x(2x1 - 0)]/(x2 - 3)2.

4⋅(x2 - 3) = 4x2 - 12
-4x(2x1 - 0) = -8x2

4x2 - 8x2 = -4x2

-(-4x2 - 12) = 4x2 + 12

So y' = [4x2 + 12]/(x2 - 3)2 is the derivative of the given function.