Abstract
The concept of the derivative of a function is introduced and formulas proved for the derivatives of sums, products, quotients and compositions of functions. The mean value theorem for derivatives is then proved and some applications explored. The ideas of a local maximum and a local minimum of a function are introduced and derivatives used to locate them. Derivatives are then applied to study further properties of the sine and cosine functions and to define inverses for them and the tangent function. Geometric interpretations are offered for the trigonometric functions. The use of derivatives to evaluate limits of functions is encapsulated in l’Hôpital’s rule, for which a discrete version is also supplied. The chapter concludes with the differentiation of power series.
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Little, C.H.C., Teo, K.L., van Brunt, B. (2015). Differentiability. In: Real Analysis via Sequences and Series. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2651-0_6
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DOI: https://doi.org/10.1007/978-1-4939-2651-0_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-2650-3
Online ISBN: 978-1-4939-2651-0
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