# Question f73f9

Feb 2, 2017

$m = - \frac{1}{10}$

#### Explanation:

color(orange)"Reminder "m_("tangent")=dy/dx#

differentiate $\textcolor{b l u e}{\text{implicitly with respect to x}}$

Both terms on the left side require to be differentiated using
the $\textcolor{b l u e}{\text{product rule}}$

$\left(x .3 {y}^{2} \frac{\mathrm{dy}}{\mathrm{dx}} + {y}^{3} .1\right) + \left(x . \frac{\mathrm{dy}}{\mathrm{dx}} + y .1\right) = 0$

$\Rightarrow 3 x {y}^{2} \frac{\mathrm{dy}}{\mathrm{dx}} + {y}^{3} + x \frac{\mathrm{dy}}{\mathrm{dx}} + y = 0$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} \left(3 x {y}^{2} + x\right) = - {y}^{3} - y$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{{y}^{3} + y}{3 x {y}^{2} + x}$

Substitute the coordinates of (5 ,1) into $\frac{\mathrm{dy}}{\mathrm{dx}}$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{2}{15 + 5} = - \frac{1}{10}$