Question

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

Answer 1

We use the chain rule.

http://socratic.org/calculus/basic-differentiation-rules/chain-rule

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 power rule, `(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)`