1. **State the problem:** Find the first derivative of the function $$f(x) = \sqrt{x^{37}}$$ with respect to $$x$$. 2. **Rewrite the function:** Recall that $$\sqrt{a} = a^{\frac{1}{2}}$$, so we rewrite the function as $$f(x) = (x^{37})^{\frac{1}{2}}$$. 3. **Simplify the exponent:** Using the power of a power rule $$ (a^m)^n = a^{mn} $$, we get $$f(x) = x^{37 \times \frac{1}{2}} = x^{\frac{37}{2}}$$. 4. **Apply the power rule for derivatives:** The derivative of $$x^n$$ is $$nx^{n-1}$$. So, $$f'(x) = \frac{37}{2} x^{\frac{37}{2} - 1} = \frac{37}{2} x^{\frac{35}{2}}$$. 5. **Rewrite the derivative with a radical:** Since $$x^{\frac{35}{2}} = x^{17 + \frac{1}{2}} = x^{17} \cdot x^{\frac{1}{2}} = x^{17} \sqrt{x}$$, the derivative is $$f'(x) = \frac{37}{2} x^{17} \sqrt{x}$$. 6. **Final answer:** $$\boxed{\frac{d}{dx} \sqrt{x^{37}} = \frac{37}{2} x^{17} \sqrt{x}}$$