We shall now investigate more systematically the forms of the derivatives of a few of the the simplest types of functions.

A. **Polynomials.** If \(\phi(x) = a_{0}x^{n} + a_{1}x^{n-1} + \dots + a_{n}\), then \[\phi'(x) = na_{0}x^{n-1} + (n – 1)a_{1}x^{n-2} + \dots + a_{n-1}.\] It is sometimes more convenient to use for the standard form of a polynomial of degree \(n\) in \(x\) what is known as the *binomial form*, viz. \[a_{0}x^{n} + \binom{n}{1} a_{1}x^{n-1} + \binom{n}{2} a_{2}x^{n-2} + \dots + a_{n}.\] In this case \[\phi'(x) = n \left\{ a_{0}x^{n-1} + \binom{n – 1}{1} a_{1}x^{n-2} + \binom{n – 1}{2} a_{2}x^{n-3} + \dots + a_{n-1} \right\}.\]

The binomial form of \(\phi(x)\) is often written symbolically as \[(a_{0}, a_{1}, \dots, a_{n} | x, 1)^{n};\] and then \[\phi'(x) = n(a_{0}, a_{1}, \dots, a_{n-1} | x, 1)^{n-1}.\]

We shall see later that \(\phi(x)\) can always be expressed as the product of \(n\) factors in the form \[\phi(x) = a_{0}(x – \alpha_{1})(x – \alpha_{2}) \dots (x – \alpha_{n}),\] where the \(\alpha\)’s are real or complex numbers. Then \[\phi'(x) = a_{0}\sum (x – \alpha_{2})(x – \alpha_{3}) \dots (x – \alpha_{n}),\] the notation implying that we form all possible products of \(n – 1\) factors, and add them all together. This form of the result holds even if several of the numbers \(\alpha\) are equal; but of course then some of the terms on the right-hand side are repeated. The reader will easily verify that if \[\phi(x) = a_{0}(x – \alpha_{1})^{m_{1}} (x – \alpha_{2})^{m_{2}}\dots (x – \alpha_{\nu})^{m_{\nu}},\] then \[\phi'(x) = a_{0} \sum m_{1}(x – \alpha_{1})^{m_{1}-1} (x – \alpha_{2})^{m_{2}}\dots (x – \alpha_{\nu})^{m_{\nu}}.\]

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