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1 3.2 The Derivative as a FunctionDefinition: The Derivative of the function ?(?)with respect to the variable ?is the function ?′is defines as ?′(?) = limℎ→0?(? + ℎ) − ?(?)ℎprovided 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 ContinuityIf ?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 FunctionDifferentiability on an Interval: One Sided Derivatives One-Sided Derivative