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4: Differentiation rules, simple antiderivatives and applications

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    In Chapter 2 we defined the derivative of a function, \(y=f(x)\) by

    \[\frac{d y}{d x}=f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} . \nonumber \]

    Using this formula, we calculated derivatives of a few power functions. Here, we gather results so far, and observe a pattern, the power rule. This rule allows us to compute higher derivatives (e.g. second derivative etc.), to differentiate polynomials, and even to find antiderivatives by applying the rule "in reverse" (finding a function that has a given derivative). All these calculations are useful in common applications, including accelerated motion. These are investigated later in this chapter. We round out the technical material by stating several other rules of differentiation (product and quotient), allowing us to easily calculate derivatives of rational functions.

    This page titled 4: Differentiation rules, simple antiderivatives and applications is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Leah Edelstein-Keshet via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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