The function f(x) can be differentiated to give its first derivative function - f'(x), and then again to give its second derivative function - f"(x ). This second derivative can be thought of as the "gradient of the gradient", and is particularly useful in deciding whether or not a stationary point of f(x), where f'(x) = 0, is:
a maximum point - f"(x) is negative
a minimum point - f"(x) is positive
f"(x) = 0 could mean a point of inflexion, or a maximum or a minimum - further investigation is necessary.
The graph of f(x) can be set as quartic: Ax⁴ + Bx³ + Cx² + Dx + E

or cubic: Bx³ + Cx² + Dx + E.
The graphs can be changed by clicking the A, B, C, D and E buttons.
By checking or unchecking the boxes at the top of the screen, you can choose whether you want to see the first derivative function - f'(x), or the second derivative function - f"(x), or both (or neither).
By moving the small circle on the x-axis from left to right you'll see how f(x), f'(x) and f"(x) are related. Play around.