Quotient Rule in Differentiation

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

[f(x)/g(x)]' = [f'(x)g(x) - f(x)g'(x)] / [g(x)]2

Proof

Quotient Rule in Differentiation: Proof of the Formula

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

Find the derivative of the given function. y = 4x/(x^2 - 3x)

y = 4x/(x2 - 3x)

y' is equal to,
the derivative of 4x, (4⋅1x0)
times (x2 - 3x)
minus (4x),
times, the derivative of x2 - 3x, (2x1 - 3⋅1x0)

over (x2 - 3x)2.

Derivatives of polynomials