1. **State the problem:** We are given two functions $f$ and $g$, and their composition $h = g \circ f$. We know $f(2) = 4$, $f'(2) = \frac{1}{2}$, and $g'(4) = 10$. We want to find $h'(2)$. 2. **Recall the chain rule:** For the composition $h(x) = g(f(x))$, the derivative is given by $$h'(x) = g'(f(x)) \cdot f'(x).$$ 3. **Apply the chain rule at $x=2$:** Substitute $x=2$ into the formula: $$h'(2) = g'(f(2)) \cdot f'(2).$$ 4. **Use the given values:** We know $f(2) = 4$, so $$h'(2) = g'(4) \cdot f'(2) = 10 \cdot \frac{1}{2}.$$ 5. **Calculate the product:** $$h'(2) = 10 \times \frac{1}{2} = 5.$$ **Final answer:** $$\boxed{5}.$$