Define the second derivative in calculus.

The second derivative is the rate of change of the first derivative with respect to the independent variable.

In calculus, the second derivative is the derivative of the first derivative of a function. It measures the rate of change of the slope of the function with respect to the independent variable. The second derivative is denoted by f''(x) or d²y/dx², where y is the function of x.

Geometrically, the second derivative represents the curvature of the graph of the function. If the second derivative is positive, the graph is concave up, and if it is negative, the graph is concave down. If the second derivative is zero, the graph has an inflection point.

The second derivative test is a method used to determine the nature of critical points of a function. If the second derivative is positive at a critical point, the function has a local minimum at that point. If the second derivative is negative, the function has a local maximum. If the second derivative is zero, the test is inconclusive.

To find the second derivative of a function, we differentiate the first derivative with respect to the independent variable. For example, if f(x) = x³ + 2x² - 5x, then f'(x) = 3x² + 4x - 5 and f''(x) = 6x + 4.