1. **State the problem:** We are given values for functions $f$ and $g$ and their derivatives at $x=7$: $f(7) = -5$, $f'(7) = 4$, $g(7) = 1$, $g'(7) = -2$. We want to find the value of the derivative of the function $h(x) = 6f(x) - 3g(x)$ at $x=7$, i.e., compute $\frac{d}{dx}(6f - 3g)(7)$. 2. **Recall the derivative rules:** The derivative of a sum/difference is the sum/difference of the derivatives. The derivative of a constant times a function is the constant times the derivative of the function. So, $$\frac{d}{dx}(6f - 3g) = 6f'(x) - 3g'(x)$$ 3. **Evaluate the derivative at $x=7$:** $$\frac{d}{dx}(6f - 3g)(7) = 6f'(7) - 3g'(7)$$ Substitute the given values: $$= 6 \times 4 - 3 \times (-2)$$ 4. **Simplify:** $$= 24 + 6 = 30$$ **Final answer:** $$\frac{d}{dx}(6f - 3g)(7) = 30$$