Part 2/6:
At its core, the derivative of a function measures the rate at which the function's output changes as its input changes. For a function denoted as ( y = f(x) ), the first derivative ( y' ) (or ( f'(x) )) represents the rate of change of the function. When taking the derivative of the first derivative, we obtain the second derivative, often written as ( y'' ) or ( f''(x) ). In mathematical terms, this can also be expressed using Leibniz notation as ( \frac{d^2y}{dx^2} ).
The derivative signifies how a function is changing, while the second derivative indicates how the rate of change itself is changing—essentially pointing to the curvature and concavity of the original function.