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.