# Question #37cdd

Jul 18, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{10 {x}^{4} + 4 {x}^{3} y}{{x}^{4} + 5 {y}^{4}}$

#### Explanation:

$2 {x}^{5} + {x}^{4} y + {y}^{5} = 36$

differentiate,
$10 {x}^{4} + {x}^{4} \frac{\mathrm{dy}}{\mathrm{dx}} + y 4 {x}^{3} + 5 {y}^{4} \frac{\mathrm{dy}}{\mathrm{dx}} = 0$

${x}^{4} \frac{\mathrm{dy}}{\mathrm{dx}} + 5 {y}^{4} \frac{\mathrm{dy}}{\mathrm{dx}} = - \left(10 {x}^{4} + 4 {x}^{3} y\right)$

$\left({x}^{4} + 5 {y}^{4}\right) \frac{\mathrm{dy}}{\mathrm{dx}} = - \left(10 {x}^{4} + 4 {x}^{3} y\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{10 {x}^{4} + 4 {x}^{3} y}{{x}^{4} + 5 {y}^{4}}$

Jul 18, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{10 {x}^{4} + 4 {x}^{3} y}{{x}^{4} + 5 {y}^{4}}$

#### Explanation:

$\text{differentiate "color(blue)"implicitly with respect to x}$

$\text{differentiate "x^4y" using the "color(blue)"product rule}$

$10 {x}^{4} + \left({x}^{4} . \frac{\mathrm{dy}}{\mathrm{dx}} + 4 {x}^{3} y\right) + 5 {y}^{4} \frac{\mathrm{dy}}{\mathrm{dx}} = 0$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} \left({x}^{4} + 5 {y}^{4}\right) = - 10 {x}^{4} - 4 {x}^{3} y$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{10 {x}^{4} + 4 {x}^{3} y}{4 {x}^{4} + 5 {y}^{4}}$