Answer
See the explanation
Work Step by Step
A second derivative represents the rate of change of the first derivative of a function with respect to its independent variable. Mathematically, if $f(x)$ is a function, then its second derivative, denoted as $f''(x)$, is the derivative of the first derivative, $f'(x)$.
Similarly, a third derivative, denoted as $f'''(x)$, represents the rate of change of the second derivative with respect to the independent variable.
The number of derivatives a function can have is theoretically infinite. The nth derivative, denoted as $f^{(n)}(x)$, represents the rate of change of the $(n-1)th$ derivative with respect to the independent variable.
For example:
- Let $f(x) = 2x^3 - 4x^2 + 7x - 1$. The first derivative is $f'(x) = 6x^2 - 8x + 7$, and the second derivative is $f''(x) = 12x - 8$.
- Taking the third derivative, $f'''(x) = 12$, which is a constant.
This demonstrates that functions can have multiple derivatives, and the process can be continued to find higher-order derivatives.
