The derivative of a function f(x) from first principles is given by the limit `lim_(h->0) (f(x+h) - f(x))/h` .

For the function `f(x) = sqrt(1+x)` , the derivative `f'(x) = lim_(h->0)(sqrt(1+x+h)-sqrt(1+x))/h`

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

= `lim_(h->0)((sqrt(1+x+h))^2-(sqrt(1+x))^2)/(h*(sqrt(1+x+h)+sqrt(1+x)))`

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

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

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

At h = 0

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

** The derivative of...**

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The derivative of a function f(x) from first principles is given by the limit `lim_(h->0) (f(x+h) - f(x))/h` .

For the function `f(x) = sqrt(1+x)` , the derivative `f'(x) = lim_(h->0)(sqrt(1+x+h)-sqrt(1+x))/h`

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

= `lim_(h->0)((sqrt(1+x+h))^2-(sqrt(1+x))^2)/(h*(sqrt(1+x+h)+sqrt(1+x)))`

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

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

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

At h = 0

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

**The derivative of `f(x) = sqrt(1+x)` is `1/(2*(sqrt(1+x)))` **