1. **State the problem:** Find the derivative of the function $f(x) = x^3 + 5x$. 2. **Recall the derivative rules:** - The derivative of $x^n$ is $nx^{n-1}$. - The derivative of a constant times a function is the constant times the derivative of the function. - The derivative of a sum is the sum of the derivatives. 3. **Apply the power rule to each term:** - Derivative of $x^3$ is $3x^{2}$. - Derivative of $5x$ is $5$. 4. **Combine the results:** $$f'(x) = 3x^{2} + 5$$ 5. **Interpretation:** The derivative $f'(x)$ gives the slope of the tangent line to the curve at any point $x$. **Final answer:** $$f'(x) = 3x^{2} + 5$$