Second Derivative

The second derivative is what you get when you differentiate a function twice: first to find the rate of change, then again to see how that rate of change itself is changing. Because it measures the slope of the slope, it tells you about the graph’s curvature: positive values usually mean the graph is concave up, negative values mean concave down. This comes directly from applying the derivative operation a second time, so it is especially useful when a function’s first derivative alone does not fully describe its shape.

Use the second derivative when you want to study concavity, identify possible inflection points, or check whether a critical point is a local maximum or minimum. It also appears in physics whenever acceleration is the second derivative of position with respect to time.

• Stopping after the first derivative and calling it the second derivative
• Forgetting to apply the power rule correctly when differentiating a second time
• Confusing concavity with the sign of the first derivative instead of the second derivative