Introduction
If
f
is a differentiable function,
f'
is its derivative.
f'
represents the slope of
f
at each point in its domain.
f'
can also have a derivative,
f''
, which represents the slope of
f'
at each point in its domain. This is called the second derivative of
f
.
Here is an example of a function and its first and second derivatives.
f(x)=\frac{1}{3}x^3
(Yellow)
f'(x)=x^2
(Red)
f''(x)=2x
(Blue)
Here is another:
f(x)=\sin(x)
(Yellow)
f'(x)=\cos(x)
(Red)
f''(x)=-\sin(x)
(Blue)
Note that our other notation
(d/dx)
can be used:
\begin{align*}
f(x) &= \sin(x) \\\\
f'(x) &= \frac{d}{dx}\sin(x)=\cos(x) \\\\
f''(x) &= \frac{d^2}{dx^2}\sin(x)=\frac{d}{dx}\cos(x)=-\sin(x) \\\\
f'''(x) &= \frac{d^3}{dx^3}\sin(x)=\frac{d}{dx}-\sin(x)=-\cos(x)
\end{align*}