Higher-order derivatives have the ability to capture information about a function that first-order derivatives alone cannot.
First-order derivatives can capture critical information like the rate of change, but they can’t tell the difference between local minima and maxima with the same rate of change.
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Several optimization techniques use the usage of higher-order derivatives to overcome this limitation, such as Newton’s approach, which uses second-order derivatives to attain the local minimum of an optimization function.
The second-order derivative is the most commonly employed derivative in machine learning. We already discussed how the second derivative can offer us information that the first derivative alone cannot.
It can inform us whether a critical point is a local minimum or maximum (depending on whether the second derivative is higher or smaller than zero), while the first derivative would otherwise be zero in both circumstances.
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