Question

find dy/dx by implicit differentiation (1+sin^3(xy^2))^1/2= y

Answer 1

Well all I have to say is chain rule

So,

Start with u = 1+sin^3(xy^2)

then d/du( sqrt(u)) = 1/(2(sqrt(u)))


then take derivative with respect to x of

1+sin^3(xy^2) = 3sin(xy^2)^2

At this point you'll have ,

(3sin(xy^2)^2)(d/dx(sin(xy^2))/2(sqrt((1+sin^3(xy^2))) = y'

then take derivative with respect to

(d/dx(sin(xy^2) = (cos(xy^2))

Then take derivative of interior and you will get

d/dx(xy^2) = y^2 + 2xyy'

So, you will have this in total

(3sin(xy^2)^2)(cos(xy^2))(y^2 + 2xyy')/2(sqrt((1+sin^3(xy^2))) = y'

I will leave it to you to solve for y' because at this point it is just algebra :)