# How do you differentiate g(z)=(z^2+1)/(x^3-5) using the quotient rule?

Sep 20, 2017

$\frac{- {z}^{4} - 3 {z}^{2} - 10 z}{{z}^{6} - 10 {z}^{3} + 25}$

#### Explanation:

The quotient rule can be stated as;

$\frac{d}{\mathrm{dz}} f \frac{z}{g} \left(z\right) = \frac{f ' \left(z\right) g \left(z\right) - f \left(z\right) g ' \left(z\right)}{g} ^ 2 \left(z\right)$

We can choose our $f \left(z\right)$ and $g \left(z\right)$ and take each derivative separately. We only need the power rule here.

$f \left(z\right) = {z}^{2} + 1$
$f ' \left(z\right) = 2 z$

$g \left(z\right) = {z}^{3} - 5$
$g ' \left(z\right) = 3 {z}^{2}$

Now we have all of the pieces we need, we can plug them into our power rule function.

$\frac{2 z \left({z}^{3} - 5\right) - 3 {z}^{2} \left({z}^{2} + 1\right)}{{z}^{3} - 5} ^ 2$

Now we can simplify our terms.

$\frac{2 {z}^{4} - 10 z - 3 {z}^{4} - 3 {z}^{2}}{{z}^{6} - 10 {z}^{3} + 25}$

$\frac{- {z}^{4} - 3 {z}^{2} - 10 z}{{z}^{6} - 10 {z}^{3} + 25}$