# Find the derivative of f(x). Use the Quotient Rule. \ f(x) = \frac{2-x^2}{3x^2+1}

## Question:

Find the derivative of {eq}f(x) {/eq}. Use the Quotient Rule. {eq}\ f(x) = \frac{2-x^2}{3x^2+1} {/eq}

## Quotient Rule

Many different rules can be applied to functions in order to find their derivatives. One rule, which is used when our function is a quotient, is the aptly named Quotient Rule.

{eq}f'(x) = \frac{g'h-gh'}{h^2} {/eq}

## Answer and Explanation:

Since this is a rational function, which is a ratio of two polynomials, its derivative must be constructed through application of the Quotient Rule. The first step in such a problem is to define the pieces that the Quotient Rule requires us to use. Since both the numerator and denominator are polynomials, their individual derivatives can be found using only the Power Rule.

{eq}\begin{align*} &g = 2-x^2 && g' = -2x\\ &h = 3x^2+1 &&h' = 6x \end{align*} {/eq}

With this information defined, we can now find the derivative of the original function by combining it together in the Quotient Rule.

{eq}\begin{align*} f'(x) &= \frac{-2x(3x^2+1) - 6x(2-x^2)}{(3x^2+1)^2}\\ &= \frac{-6x^3-2x-12x+6x^3}{(3x^2+1)^2}\\ &= \frac{-14x}{(3x^2+1)^2}\\ &= -\frac{14x}{(3x^2+1)^2} \end{align*} {/eq}