Find the derivative of #x+sqrtx# using the definition of derivative?

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Answer 1 · 2017-04-14T14:30:05

#d/(dx)(x+sqrtx)=1+1/(2sqrtx)#

#(df)/(dx)# is defined as #Lt_(h->0)(f(x+h)-f(x))/h#

Here #f(x)=x+sqrtx# and hence

#f(x+h)=x+h+sqrt(x+h)# and

#f(x+h)-f(x)=h+sqrt(x+h)-sqrtx# and hence

#(df)/(dx)=Lt_(h->0)[1+(sqrt(x+h)-sqrtx)/h]#

= #1+Lt_(h->0)((sqrt(x+h)-sqrtx)(sqrt(x+h)+sqrtx))/(h(sqrt(x+h)+sqrtx))#

= #1+Lt_(h->0)h/(h(sqrt(x+h)+sqrtx))#

= #1+Lt_(h->0)1/(sqrt(x+h)+sqrtx)#

= #1+1/(2sqrtx)#