How do you find f'(x) using the definition of a derivative for #f(x)=x + sqrtx + 7#?

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Answer 1 · 2015-09-27T04:27:43

#dy/dx=1+1/(2sqrtx)#

#f(x)=x+sqrtx+7#
#dy/dx=1+1/(2sqrtx)#

Answer 2 · 2015-09-27T04:53:35

See the explanation.

#f'(x)=lim_(h->0) (f(x+h)-f(x))/h#

#lim_(h->0) (x+h+sqrt(x+h)+7-x-sqrtx-7)/h=#

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

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

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

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

#=1+1/(sqrtx+sqrtx)=1+1/(2sqrtx)#