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Using the alternate definition of the derivative

Using the alternate definition of the derivative \(\left(f^{\prime}(c)=\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}\right)\), find \(f^{\prime}(c)\) where \(f(x)=2 x^{3} .\)

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Given f(x)=2x² F'c= lim fluj -f(c) ac n-c 2x3 - 23 - lim k- (x-2) alim KAC 2 (x2- C3) (x-C) Apply glade adentity : a3_b² = (a-bla + b + ab) lim 2 (x7CKP 2x) ntc (x2+c²7 (a- = lim a cm2 +62+2x) 7c lim 27c 2x²tqc2 t und Apply limit FC) = = 2 c² +9c² + 4cxc 4c2 +42 8c2 fico)

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