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Higher derivatives

So far in this module, we have looked at first and second derivatives. But there's no need to stop differentiating after doing it twice!

Given a function \(f(x)\), if it can be differentiated \(k\) times, then we write the \(k\)th derivative as \(f^{(k)}(x)\). In Leibniz notation, the \(k\)th derivative of \(y\) with respect to \(x\) is denoted by \(\dfrac{d^k y}{dx^k}\).

Higher derivatives are useful for finding polynomial approximations to functions, as we see in the next section.

Exercise 24

Let \(n\) be a positive integer. What is the \(n\)th derivative of the function \(f(x) = x^n\)?

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