1. **State the problem:** We are given two functions $f(x) = 5 - x^3$ and $g(x) = x^2$. We need to find the derivative of the composite function $(g \circ f)'(x)$, which means the derivative of $g(f(x))$. 2. **Recall the chain rule:** The derivative of a composite function $g(f(x))$ is given by: $$ (g \circ f)'(x) = g'(f(x)) \cdot f'(x) $$ This means we first find the derivative of the outer function $g$ evaluated at $f(x)$, then multiply by the derivative of the inner function $f$. 3. **Find $f'(x)$:** Given $f(x) = 5 - x^3$, differentiate term-by-term: $$ f'(x) = 0 - 3x^2 = -3x^2 $$ 4. **Find $g'(x)$:** Given $g(x) = x^2$, differentiate: $$ g'(x) = 2x $$ 5. **Apply the chain rule:** Substitute $f(x)$ into $g'(x)$: $$ g'(f(x)) = 2(5 - x^3) $$ 6. **Multiply by $f'(x)$:** $$ (g \circ f)'(x) = 2(5 - x^3) \cdot (-3x^2) $$ 7. **Simplify:** $$ (g \circ f)'(x) = -6x^2 (5 - x^3) = -30x^2 + 6x^5 $$ **Final answer:** $$ (g \circ f)'(x) = -30x^2 + 6x^5 $$