## Calculus Tutorial

#### Intro

The quotient rule is when a larger function is composed of two differentiable functions divided by one another like so:

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

#### Sample Problem

Find the first derivative of the following function:

$f(x) = \frac{2x^2 + 3}{5x^3}$

#### Solution

f(x) can be broken down where:

$g(x) = 2x^3 + 3$

$h(x) = 5x^3$

and their individual derivatives are:

$g'(x) = 4x$

$h'(x) = 15x^2$

Now plug into the quotient rule formula:

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

$\frac{(4x)(5x^3) - (2x^2 + 3)(15x^2)}{(5x^3)^2} = \frac{20x^4 - 30x^4 - 45x^2}{25x^6} = \frac{-10x^4 - 45x^2}{25x^6}$

Reduce further for the final answer:

$\frac{-2x^2 - 9}{5x^4}$