How do you use the limit definition of the derivative to find the derivative of #f(x)=sqrtx#?

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Answer 1 · 2016-09-17T06:32:46

# f'(x)=1/(2sqrtx)#.

By defn., #f'(x)=lim_(trarrx) (f(t)-f(x))/(t-x)#.

For #f(x)=sqrtx=x^(1/2), f'(x)=lim_(trarrx) (t^(1/2)-x^(1/2))/(t-x)#

#=lim_(trarrx) (t^(1/2)-x^(1/2))/{(t^(1/2))^2-(x^(1/2))^2}#

#=lim_(trarrx) cancel((t^(1/2)-x^(1/2)))/{(t^(1/2)+x^(1/2))cancel((t^(1/2)-x^(1/2))}#

#=lim_(trarrx) 1/(t^(1/2)+x^(1/2)#

#=1/(x^(1/2)+x^(1/2))#

#=1/(2x^(1/2))#

#:. f'(x)=1/(2sqrtx)#.