How do you derive y=tanx using the definition of the derivative?

Use the tangent of a sum, continuity of tangent and ##lim_(hrarr0)tan h/h = 1##

##f(x) = tanx##

##f'(x) = lim_(hrarr0)(tan(x+h) - tanx)/h##

## = lim_(hrarr0)((tanx+tan h)/(1-tanxtan h) - tanx)/h##

## = lim_(hrarr0)(tanx+tan h- tanx+tan^2xtan h)/(h(1-tanxtan h)) ##

## = lim_(hrarr0)(tan h/h * (1+tan^2x)/(1-tanxtan h)) ##

## = (1) * (1+tan^2x)/(1-0) = 1+tan^2x = sec^2x##