ExampleLetf(x) =√3x .Use the definition of the derivative to findf0(3) = limh→0f(3 +h)-f(3)h,(1)and then write the equation of the tangent line to the curvey=f(x) at the point(3, f(3)).Solution:Making the substitutionf(3 +h) =q3(3 +h) into (1), it follows thatf0(3)=limh→0q3(3 +h)-√3·3h=limh→0√9 + 3h-3h.Multiplying numerator and denominator in the above by the conjugate expression,√9 + 3h+ 3,to√9 + 3h-3, we havef0(3)=limh→0√9 + 3h-3h·√9 + 3h+ 3√9 + 3h+ 3=limh→0(9 + 3h)-9h√9 + 3h+ 3=limh→03hh√9 + 3h+ 3=limh→03√
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