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

1 Answer
Psykolord1989 . ยท Truong-Son N.
May 29, 2015

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)

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