# What is the implicit derivative of 13=2xe^y-x^3e^(2y) ?

Dec 1, 2015

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 {x}^{2} {e}^{y} - 2}{2 x \left(1 - {x}^{2} {e}^{y}\right)}$

#### Explanation:

This will require Implicit Differentiation, Chain Rule, and Product Rule. The most important thing to remember is that since we're differentiating with respect to $x$, anything with $y$ will spit out a $\frac{\mathrm{dy}}{\mathrm{dx}}$ term.

$\frac{d}{\mathrm{dx}} \left[13 = 2 x {e}^{y} - {x}^{3} {e}^{2 y}\right]$

Let's first find the derivative of each inner part.

$\frac{d}{\mathrm{dx}} \left[2 x {e}^{y}\right] = 2 \frac{d}{\mathrm{dx}} \left[x {e}^{y}\right] = 2 \left({e}^{y} + x {e}^{y} \frac{\mathrm{dy}}{\mathrm{dx}}\right) = 2 {e}^{y} + 2 x {e}^{y} \frac{\mathrm{dy}}{\mathrm{dx}}$

$\frac{d}{\mathrm{dx}} \left[{x}^{3} {e}^{2 y}\right] = 3 {x}^{2} {e}^{2 y} + 2 {x}^{3} {e}^{2 y} \frac{\mathrm{dy}}{\mathrm{dx}}$

Back to the original:

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

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

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

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

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 {x}^{2} {e}^{y} - 2}{2 x \left(1 - {x}^{2} {e}^{y}\right)}$