Second Derivatives via Formulas - At A Glance

When we take the derivative of a differentiable function f, we end with a new function f '.
If this new function f ' is differentiable, then we can take its derivative to find (f ')', also known as f " or the second derivative of f.

Sample Problem

If f (x) = x² + 4x, then we take its derivative once to find

f '(x) = 2x + 4.

Since 2x + 4 is a differentiable function, we can take its derivative to find

f "(x) = 2.

Sample Problem

If f (x) = cos x then f '(x) = -sin x. The second derivative of f is the derivative of -sin x, so
f "(x) = -cos x.

Not all continuous functions are differentiable, and not all differentiable functions have second derivatives.

Example. Look at the piecewise-defined function

This function is continuous, since it is continuous on each of the pieces and since

It is also differentiable. We can take the derivative of each piece to find

This new function f ' is the absolute value function. We could write

f '(x) = |x|.

We know that the absolute value function is not differentiable, which means we can't take another derivative here. The second derivative of f, also known as f "(x), does not exist.