Second Derivatives and Beyond
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.
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.
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.