# 3.2 The Derivative as a Function.pdf - 1 3.2 The Derivative...

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1 3.2 The Derivative as a Function Definition: The Derivative of the function ?(?) with respect to the variable ? is the function ?′ is defines as ? (?) = lim ℎ→0 ?(? + ℎ) − ?(?) provided this limit exists. Alternate Formula of ?′ ? (?) = lim ? → ? ?(?) = ?(?) ? − ? The process of finding a derivative is called Differentiation. Notation: ? ( ?) is also denoted as ?? ?? When the derivative ?(? 0 ) exists, we say ? is differentiable at ? 𝟎 . If a function ? is differentiable at each number in its domain, we say ? 𝒊? ? ?𝒊???????𝒊??𝒍? ?????𝒊?? . Theorem: Differentiability Implies Continuity If ? has a derivative at ? = ? , then ? is continuous at ? = ? . Calculating the Derivative Using the Definition Example 1:Find the derivative of ?(?) = 4 − 3?2using the definition of the derivative. Calculate ?(−2),?(0),and ?′(3). 𝑑? [
2 3.2 The Derivative as a Function Differentiability on an Interval: One Sided Derivatives One-Sided Derivative