1. The problem is to determine when to use the first derivative or the second derivative in calculus. 2. The first derivative of a function $f(x)$, denoted as $f'(x)$, represents the rate of change or slope of the function at any point $x$. 3. The second derivative, denoted as $f''(x)$, is the derivative of the first derivative and represents the curvature or concavity of the function. 4. Use the first derivative to find: - Increasing or decreasing behavior of the function. - Critical points where $f'(x) = 0$ or is undefined, which may indicate local maxima, minima, or saddle points. 5. Use the second derivative to determine: - Concavity of the function: if $f''(x) > 0$, the function is concave up; if $f''(x) < 0$, it is concave down. - Points of inflection where the concavity changes, found by solving $f''(x) = 0$. 6. Summary: - Use the first derivative to analyze slope and find critical points. - Use the second derivative to analyze concavity and points of inflection. This helps in understanding the shape and behavior of the graph of a function.