Derivative of a product of two composite functions

First apply the product rule with $u = (5x-1)^3$ and $v = (2x+3)^2$, so $f' = u'v + uv'$. To find $u'$, use the chain rule: $u' = 3(5x-1)^2 \cdot 5 = 15(5x-1)^2$. To find $v'$, use the chain rule: $v' = 2(2x+3) \cdot 2 = 4(2x+3)$.

Substituting into the product rule gives $f'(x) = 15(5x-1)^2(2x+3)^2 + (5x-1)^3 \cdot 4(2x+3)$. Factoring out the common factor $(5x-1)^2(2x+3)$ yields $f'(x) = (5x-1)^2(2x+3)\left[15(2x+3) + 4(5x-1)\right]$, which simplifies to $f'(x) = (5x-1)^2(2x+3)(50x + 41)$.