# How do you use implicit differentiation to find (dy)/(dx) given 5x^3+xy^2=5x^3y^3?

Jan 22, 2017
1. Differentiate each term
$d \left(5 {x}^{3}\right) + d \left(x {y}^{2}\right) = d \left(5 {x}^{3} {y}^{3}\right)$

2. Pull constants out of each differential
$5 d \left({x}^{3}\right) + d \left(x {y}^{2}\right) = 5 d \left({x}^{3} {y}^{3}\right)$

3. Differentiate
$5 \left(\textcolor{red}{3 {x}^{2} \mathrm{dx}}\right) + \left[\textcolor{b l u e}{x \left(2 y\right) \mathrm{dy} + {y}^{2} \mathrm{dx}}\right] = 5 \left[\textcolor{p u r p \le}{{x}^{3} \left(3 {y}^{2}\right) \mathrm{dy} + {y}^{3} \left(3 {x}^{2}\right) \mathrm{dx}}\right]$

4. Simplify
$\left(15 {x}^{2}\right) \mathrm{dx} + \left(2 x y\right) \mathrm{dy} + \left({y}^{2}\right) \mathrm{dx} = \left(15 {x}^{3} {y}^{2}\right) \mathrm{dy} + \left(15 {x}^{2} {y}^{3}\right) \mathrm{dx}$

5. Separate $\mathrm{dy}$ and $\mathrm{dx}$ terms
$\left(2 x y\right) \mathrm{dy} - \left(15 {x}^{3} {y}^{2}\right) \mathrm{dy} = \left(15 {x}^{2} {y}^{3}\right) \mathrm{dx} - \left(15 {x}^{2}\right) \mathrm{dx} - \left({y}^{2}\right) \mathrm{dx}$

6. Simplify terms
$\left(- 15 {x}^{3} {y}^{2} + 2 x y\right) \mathrm{dy} = \left(15 {x}^{2} {y}^{3} - 15 {x}^{2} - {y}^{2}\right) \mathrm{dx}$

7. Solve for $\frac{\mathrm{dy}}{\mathrm{dx}}$
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{15 {x}^{2} {y}^{3} - 15 {x}^{2} - {y}^{2}}{- 15 {x}^{3} {y}^{2} + 2 x y}$