Let’s consider the function

\[y = f(x) = x^5 - 3x^4 + x\]

Its derivative \(f^\prime\) is

\[y^\prime = f^\prime(x) = 5x^4 - 12x^3 + 1\]

The derivative function \(f^\prime\) can also be differientiated. We can think of the derivative of \(f^\prime\) as rhe rate of change of the slope of the tange lines of \(f\). We use the notation \(f^{\prime\prime}\).

\[f^{\prime\prime}(x) = \frac{d}{dx}f^\prime(x)\]

We call this function the second derivative of \(f\). Following this reasoning we can expand to \(f^{\prime\prime\prime}\) or \(f^{(3)}(x)\), \(f^{\prime\prime\prime\prime}\) or \(f^{(4)}(x)\), and so on. Hence, higher-order derivatives.