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**Junesong** This is a homework problem on the Algebra of Derivatives section in my text. I'm not sure if I'm handling the $\displaystyle f(x^3)$ part correctly. Any suggestions (or if it's right, let me know) would help ease my mind greatly.

__Problem__

$\displaystyle f:R \rightarrow R$ is differentiable.

$\displaystyle g(x)=x^2 f(x^3)$

Prove $\displaystyle g(x)$ is differentiable and find the derivative.

__Proof__

We know that $\displaystyle f(x)$ is differentiable over the real numbers, and clearly, if $\displaystyle x$ is a real number, then $\displaystyle x^3$ is a real number, so $\displaystyle f(x^3)$ is differentiable. Let $\displaystyle h(x)=x^2$. Then $\displaystyle h(x)$ is differentiable, and by the Algebra of Derivatives, $\displaystyle g(x)$ must be differentiable.

It follows from the product rule that $\displaystyle g(x)=2xf(x^3)+x^2 f'(x^3)$.