# Using first principles determine the derivative of f(x) = sqrt x

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

Here `f(x) = sqrt x`

The derivative in this case is:

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

Multiply the numerator and denominator by `sqrt(x + h) + sqrt x` .

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

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

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

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

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

Substitute h = 0

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

= `1/(2*sqrt x)`

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

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