Find dy/dx by implicit differentiation. 4 + 7x = sin (xy^2)

Take the derivative with respect to x, where y is a function of x

Do a chain rule products rule combo

7 = y²Cos (xy²) + 2xyCos (xy²) (dy/dx)

solve for dy/dx

(7 - y²Cos (xy²)) / (2xyCos (xy²)) = dy/dx

simplify

7Sec (xy²) / 2xy - y/2x = dy/dx