1. **State the problem:** We are given two functions $$f(x) = \sqrt{x}$$ and $$g(x) = x^2 + 7$$ and asked to find the derivative of the composition $$(f \circ g)'(3)$$. 2. **Recall the chain rule:** For functions $f$ and $g$, the derivative of their composition is: $$ (f \circ g)'(x) = f'(g(x)) \cdot g'(x) $$ 3. **Find $f'(x)$:** Since $$f(x) = \sqrt{x} = x^{1/2}$$, the derivative is $$f'(x) = \frac{1}{2} x^{-1/2} = \frac{1}{2 \sqrt{x}}$$ 4. **Find $g'(x)$:** Given $$g(x) = x^2 + 7$$, the derivative is $$g'(x) = 2x$$ 5. **Evaluate $f'(g(3))$:** Calculate $$g(3)$$ first: $$g(3) = 3^2 + 7 = 9 + 7 = 16$$ Then, $$f'(g(3)) = f'(16) = \frac{1}{2 \sqrt{16}} = \frac{1}{2 \times 4} = \frac{1}{8}$$ 6. **Evaluate $g'(3)$:** $$g'(3) = 2 \times 3 = 6$$ 7. **Calculate $(f \circ g)'(3)$:** $$ (f \circ g)'(3) = f'(g(3)) \times g'(3) = \frac{1}{8} \times 6 = \frac{6}{8} = \frac{3}{4} $$ 8. **Answer:** The value of $$(f \circ g)'(3)$$ is $$\frac{3}{4}$$ which corresponds to option (a).