How do you use implicit differentiation to find (dy)/(dx) given 1=3x+2x^2y^2?

Dec 24, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{- 3 - 4 x {y}^{2}}{4 {x}^{2} y}$

Explanation:

Differentiate the equation through term by term. Remember you are differentiating with respect to $x$ so when differentiating $x$ we get:

$x \to 1$

but when differentiating with respect to $y$ we get:

$y \to \frac{\mathrm{dy}}{\mathrm{dx}}$.

So:

$1 = 3 x + 2 {x}^{2} {y}^{2}$

Differentiating gives:

$0 = 3 + 4 x {y}^{2} + 4 {x}^{2} y \frac{\mathrm{dy}}{\mathrm{dx}}$

which we obtain using the chain and product rule on the $2 {x}^{2} {y}^{2}$ term. So we now re-arrange to get $\frac{\mathrm{dy}}{\mathrm{dx}}$:

$\to \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{- 3 - 4 x {y}^{2}}{4 {x}^{2} y}$