Product Rule - Applications

Applications

Among the applications of the product rule is a proof that

when n is a positive integer (this rule is true even if n is not positive or is not an integer, but the proof of that must rely on other methods). The proof is by mathematical induction on the exponent n. If n = 0 then xn is constant and nxn − 1 = 0. The rule holds in that case because the derivative of a constant function is 0. If the rule holds for any particular exponent n, then for the next value, n + 1, we have

\begin{align} {d \over dx}x^{n+1} &{}= {d \over dx}\left( x^n\cdot x\right) \\ &{}= x{d \over dx} x^n + x^n{d \over dx}x \qquad\mbox{(the product rule is used here)} \\ &{}= x\left(nx^{n-1}\right) + x^n\cdot 1\qquad\mbox{(the induction hypothesis is used here)} \\ &{}= (n + 1)x^n. \end{align}

Therefore if the proposition is true of n, it is true also of n + 1.

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