# How do you implicitly differentiate 2=(x+2y)^2-xy-e^(3x+y^2) ?

$y ' = \frac{2 x + 3 y - 3 \cdot {e}^{\left(3 x + {y}^{2}\right)}}{- 3 x - 8 y + 2 y \cdot {e}^{\left(3 x + {y}^{2}\right)}}$

#### Explanation:

From the given differentiate each term of both sides with respect to x

$2 = {\left(x + 2 y\right)}^{2} - x y - {e}^{3 x + {y}^{2}}$

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

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

$0 = 2 {\left(x + 2 y\right)}^{1} \cdot \left(1 + 2 y '\right) - x y ' - y \cdot 1 - {e}^{3 x + {y}^{2}} \cdot \left(3 + 2 y y '\right)$

$0 = \left(2 x + 4 y\right) \left(1 + 2 y '\right) - x y ' - y - {e}^{3 x + {y}^{2}} \left(3 + 2 y y '\right)$

Expand then simplify

$0 = 2 x + 4 y + 4 x y ' + 8 y y ' - x y ' - y - 3 {e}^{3 x + {y}^{2}} - 2 y y ' {e}^{3 x + {y}^{2}}$

Transpose those terms with y' to the left of the equation

$- 4 x y ' - 8 y y ' + x y ' + 2 y y ' {e}^{3 x + {y}^{2}} = 2 x + 4 y - y - 3 {e}^{3 x + {y}^{2}}$

factor out the $y '$

$\left(- 4 x - 8 y + x + 2 y {e}^{3 x + {y}^{2}}\right) y ' = 2 x + 4 y - y - 3 {e}^{3 x + {y}^{2}}$

simplify

$\left(- 3 x - 8 y + 2 y {e}^{3 x + {y}^{2}}\right) y ' = 2 x + 3 y - 3 {e}^{3 x + {y}^{2}}$

divide both sides by $\left(- 3 x - 8 y + 2 y {e}^{3 x + {y}^{2}}\right)$

$\left(- 3 x - 8 y + 2 y {e}^{3 x + {y}^{2}}\right) y ' = 2 x + 3 y - 3 {e}^{3 x + {y}^{2}}$

$\frac{\cancel{\left(- 3 x - 8 y + 2 y {e}^{3 x + {y}^{2}}\right) y '}}{\cancel{\left(- 3 x - 8 y + 2 y {e}^{3 x + {y}^{2}}\right)}}$

$= \frac{2 x + 3 y - 3 {e}^{3 x + {y}^{2}}}{- 3 x - 8 y + 2 y {e}^{3 x + {y}^{2}}}$

and

$y ' = \frac{2 x + 3 y - 3 {e}^{3 x + {y}^{2}}}{- 3 x - 8 y + 2 y {e}^{3 x + {y}^{2}}}$

God bless....I hope the explanation is useful.