How do you find the derivative of ##sin(x^3)##?

Using the notation provided there, if we define ##y(x) = sin(x^3)## and ##u=x^3##, we may rewrite ##y(x)## as ##y(u)=sin(u)##

From the chain rule we know that ##dy/dx = (du)/dx (dy)/(du)##. Recall that ##u(x) = x^3## and ##y(u) = sin(u)##. Therefore, by the , ##(du)/dx = 3x^2##, and by the definitions of trigonometric derivatives, ##dy/(du) = cos(u)##. Thus:

##dy/dx = (3x^2)(cos (u)) ##

Substituting ##x^3## back for u yields:

##dy/dx = 3x^2 cos(x^3)##