Question

How do you find f'(x) using the definition of a derivative for `f(x)= 6 x + 2sqrt{x}`?

Answer 1

The derivative is

#f'(x)=lim_(h->0)(f(x+h)-f(x))/h=lim_(h->0)(6(x+h)+2sqrt(x+h)-6x-2sqrtx)/h=lim_(h->0) 6+2*(sqrt(x+h)-sqrtx)/h= 6+2lim_(h->0)(sqrt(x+h)-sqrtx)/h= 6+2lim_(h->0)h/(h(sqrt(x+h)+sqrtx))=6+2*1/(2sqrtx)=6+1/sqrtx#