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Section2.5Proofs of the arithmetic of derivatives

The theorems of the previous section are not too difficult to prove from the definition of the derivative (which we know) and the arithmetic of limits (which we also know). In this section we show how to construct these rules.

Throughout this section we will use our two functions \(f(x)\) and \(g(x)\text{.}\) Since the theorems we are going to prove all express derivatives of linear combinations, products and quotients in terms of \(f,g\) and their derivatives, it is helpful to recall the definitions of the derivatives of \(f\) and \(g\text{:}\)

\begin{align*} f'(x) &=\lim_{h\to0} \frac{f(x+h)-f(x)}{h} &\text{and}&& g'(x) &=\lim_{h\to0} \frac{g(x+h)-g(x)}{h}. \end{align*}

Our proofs, roughly speaking, involve doing algebraic manipulations to uncover the expressions that look like the above.