# Find dy/dx by implicit differentiation for sqrt(5x + y)= 3 + x^2y^2 ?

Jul 17, 2018

$y ' = \frac{4 x {y}^{2} \sqrt{5 x + y} - 5}{1 - {x}^{2} y \sqrt{5 x + y}}$

#### Explanation:

Using the chain rule by assuming $y = y \left(x\right)$ we get

$\frac{1}{2} {\left(5 x + y\right)}^{- \frac{1}{2}} \left(5 + y '\right) = 2 x {y}^{2} + 2 {x}^{2} y y '$

multiplying both sides by $2 \sqrt{5 x + y}$ we obtain

$5 + y ' = 4 x {y}^{2} \sqrt{5 x + y} + 4 {x}^{2} y y ' \sqrt{5 x + y}$
solving for $y '$

$y ' \left(1 - 4 {x}^{2} y \sqrt{5 x + y}\right) = 4 {x}^{2} y \sqrt{5 x + y} - 5$

so

$y ' = \frac{4 x {y}^{2} \sqrt{5 x + y} - 5}{1 - 4 {x}^{2} y \sqrt{5 x + y}}$

if $1 - 4 {x}^{2} y \sqrt{5 x + y} \ne 0$