Derivative of a Rational Function

To differentiate a rational function, we apply the quotient rule for the derivative of two functions.

What’s a rational function? It’s a function expressed as the ratio of two polynomials. For example: $$ f(x) = \frac{2x^2+3x-1}{4x^3-2x^2+3} $$

A practical example

Let’s look at the following rational function:

$$ P(x) = \frac{-x^2-2x}{1-x} $$

$$ \frac{f(x)}{g(x)} = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} $$

We’ll apply this rule to our function, where:

$$ f(x) = -x^2 - 2x $$

$$ g(x) = 1 - x $$

So we get:

$$ P'(x) = \frac{D[-x^2-2x] \cdot (1-x) - (-x^2-2x) \cdot D[1-x]}{(1-x)^2} $$

$$ P'(x) = \frac{(-2x-2) \cdot (1-x) - (-x^2-2x) \cdot (-1)}{(1-x)^2} $$

$$ P'(x) = \frac{\big(-2x + 2x^2 - 2 + 2x\big) - \big(x^2 + 2x\big)}{(1-x)^2} $$

$$ P'(x) = \frac{x^2 - 2x - 2}{(1-x)^2} $$