Understand and Use the Notation d^2y / dx^2

Understanding Second Derivative Notation d^2y/dx^2

The notation of second derivative, d^2y/dx^2, is simply differentiation applied twice. This is also known as ‘double differentiation’ or ‘second-order differentiation’.
If a function, y = f(x), is differentiated once to get f ‘(x), or dy/dx, differentiating again will give you the second derivative, f ‘‘(x), or d^2y/dx^2.

Implications of the Second Derivative

d^2y/dx^2 provides information about the ‘concavity’ (or curvature) of a function’s graph.
If d^2y/dx^2 > 0, the graph of the function is concave up, implying the function’s rate of increase is growing.
If d^2y/dx^2 < 0, the graph of the function is concave down, implying the function’s rate of increase is decreasing.
The second derivative can also be used to determine inflexion points. These are points where the graph of the function switches from being concave up to concave down, or vice versa.

Applying the Second Derivative

To apply d^2y/dx^2, simply take the first derivative dy/dx and differentiate it again with respect to x.
The rules for differentiating such as the chain rule, the product rule, and the quotient rule, still apply when finding the second derivative.

Remember, being able to find and interpret the second derivative is a key skill in calculus which can help predict and analyse the behaviour of complex functions.