Question

Find dy/dx by implicit differentiation cos(xy²)= y

Answer 1

Given

$\cos(xy^2) = y$

Chain rule:

$\dfrac{d\cos(xy^2)}{dx} = \dfrac{dy}{dx}$

$-\sin(xy^2) \dfrac{d(xy^2)}{dx} = \dfrac{dy}{dx}$

Product rule:

$-\sin(xy^2) \left(x\dfrac{dy^2}{dx} + y^2 \dfrac{dx}{dx}\right) = \dfrac{dy}{dx}$

$-\sin(xy^2) \left(2xy\dfrac{dy}{dx} + y^2\right) = \dfrac{dy}{dx}$

Solve for $\frac{dy}{dx}$.

$-2xy\sin(xy^2) \dfrac{dy}{dx} - y^2 \sin(xy^2) = \dfrac{dy}{dx}$

$\left(1+2xy\sin(xy^2)\right) \dfrac{dy}{dx} = -y^2 \sin(xy^2)$

$\dfrac{dy}{dx} = \boxed{-\dfrac{y^2 \sin(xy^2)}{1 + 2xy \sin(xy^2)}}$