Chain rule with a sine composite function

Write the function as a composition $f(u) = \sin(u)$ where $u = 5x^2 + 1$. The derivative of the outer function is $f'(u) = \cos(u)$, and the derivative of the inner function is $u' = \dfrac{d}{dx}[5x^2+1] = 10x$.

By the chain rule, $g'(x) = f'(u)\cdot u' = \cos(5x^2+1)\cdot 10x$. Rearranging gives the final answer $g'(x) = 10x\cos(5x^2+1)$.

The outer function was f of u equals sine of u. This is the function applied last, wrapping around the inner expression five x squared plus one. Recognizing this split is what allows the chain rule to be applied correctly.