Question

Find the derivative using first principles? : `x+sqrt(x)`

Answer 1

`d/dx (x+sqrt(x)) = 1+(1)/(2sqrt(x)`

Explanation:

Let us define:

`f(x) = x+sqrt(x)`

Then using the limit definition of the derivative, we have:

`f'(x) = lim_(h rarr 0) (f(x+h)-f(x))/h`

`\ \ \ \ \ \ \ \ \ = lim_(h rarr 0) ((x+h)+sqrt(x+h)-x+sqrt(x))/h`

`\ \ \ \ \ \ \ \ \ = lim_(h rarr 0) (h+sqrt(x+h)-sqrt(x))/h`

`\ \ \ \ \ \ \ \ \ = lim_(h rarr 0) 1+(sqrt(x+h)-sqrt(x))/h * (sqrt(x+h)+sqrt(x))/(sqrt(x+h)+sqrt(x))`

`\ \ \ \ \ \ \ \ \ = lim_(h rarr 0) 1+((x+h)-(x))/(h(sqrt(x+h)+sqrt(x))`

`\ \ \ \ \ \ \ \ \ = lim_(h rarr 0) 1+(h)/(h(sqrt(x+h)+sqrt(x))`

`\ \ \ \ \ \ \ \ \ = lim_(h rarr 0) 1+(1)/(sqrt(x+h)+sqrt(x)`

`\ \ \ \ \ \ \ \ \ = 1+(1)/(sqrt(x+0)+sqrt(x)`

`\ \ \ \ \ \ \ \ \ = 1+(1)/(2sqrt(x)`