Determine the derivative of sqrt x using first principles.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The derivative of a function f(x) is given by `f'(x) = lim_(h->0) (f(x+h) - f(x))/h`

For `f(x) = sqrt x`

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

substituting h = 0 gives the indeterminate form `0/0` , multiply the numerator and denominator by `(sqrt(x + h) + sqrt x)`

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

Use the relation `(a - b)(a +b) = a^2 - b^2`

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

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

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

substituting h = 0

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

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

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