Chain rule with a polynomial inner function

Identify the outer function as $f(u) = u^4$ and the inner function as $u = g(x) = 3x^2 - 5x + 1$. By the chain rule, $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$, where $f'(u) = 4u^3$.

Compute the derivative of the inner function: $g'(x) = 6x - 5$. Substituting back gives $f'(x) = 4(3x^2 - 5x + 1)^3 \cdot (6x - 5)$, which is the final simplified derivative.