# 3.1 Tangent Lines and Derivative at a Point.pdf - 3.1...

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3.1 Tangent Lines and Derivative at a Point 1 of 2 In chapter 2, we discussed average rate of change and instantaneous rate of change Average rate of change of 𝑓 between ? 0 and ? 0 + ℎ is 𝑓(𝑥 0 +ℎ)−𝑓(𝑥 0 ) Instantaneous rate of change of 𝑓 at ? 0 is lim ℎ→0 𝑓(𝑥 0 +ℎ)−𝑓(𝑥 0 ) Average rate of change = slope of the secant line through (? 0 , 𝑓(? 0 ) and (? 0 + ℎ, 𝑓(? 0 + ℎ) . Instantaneous rate of change is equal to rate of change when ? = ? 0 . slope of the tangent line at the point (? 0 , 𝑓(? 0 )) . The slope of the curve ? = 𝑓(?) at the point (? 0 , 𝑓(? 0 )) . In this section, the instantaneous rate of change at ? 0 will be called the derivative of 𝒇 at 𝒙 𝟎 . Definition: The derivative of a function 𝒇 at a point 𝒙 𝟎 , denoted by 𝒇′(𝒙 𝟎 ) is 𝑓 (? 0 ) = lim ℎ→0 𝑓(? 0 + ℎ) − 𝑓(? 0 ) provided this limit exists. Note: ℎ ≠ 0 . Example 1: Find the derivative of 𝑓(?) = ? 2 + 5 at ? = 3 . Solution: