# How do find the second derivative for dy/dx = 5x^2 - 6/(y-2)?

Refer to explanation

#### Explanation:

We know that $y ' = \frac{\mathrm{dy}}{\mathrm{dx}} = 5 {x}^{2} - \frac{6}{y - 2}$ hence

$\frac{{d}^{2} y}{{d}^{2} x} = \frac{d}{\mathrm{dx}} \left(5 {x}^{2} - 6 \cdot {\left(y - 2\right)}^{- 1}\right) = 25 x + 6 {\left(y - 2\right)}^{- 2} \cdot \left(y\right) '$

But replacing the value of the first derivative we have that

$\frac{{d}^{2} y}{{d}^{2} x} = 25 x + 6 {\left(y - 2\right)}^{- 2} \cdot \left(5 {x}^{2} - \frac{6}{y - 2}\right) = 25 x + \frac{30 {x}^{2}}{{\left(y - 2\right)}^{2}} - \frac{36}{{\left(y - 2\right)}^{3}}$