Implicitly Defined Functions

Question

Consider the implicitly defined function 1+x=sin(xy²). Find dy/dx.

Answer 1

Differentiate:

1=cos(xy²)(2xy(dy/dx)+y²)

sec(xy²)=2xy(dy/dx)+y²

sec(xy²)-y²=2xy(dy/dx)

You finish --- one single division step

Answer 2

1+x=sin(xy²)

1=cos(xy²)(x •2ydy/dx + y² •1) [chain rule]

1=cos(xy²)(2xy dy/dx + y²)

1=2xycos(xy²) dy/dx + y²cos(xy²)

dy/dx=1 - y²cos(xy²) / 2xycos(xy²)