Use the limit definition of derivatives to differentiate `f(x) = sqrt(8x)`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The derivative of `f(x) = sqrt (8x)` has to be determined using first principles.

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

=> `lim_(h->0) (sqrt(8x + 8h) - sqrt (8x))/h`

=> `lim_(h->0)( (sqrt(8x + 8h) - sqrt (8x))(sqrt(8x+8h)+sqrt (8x)))/(h*(sqrt(8x + 8h) + sqrt (8x)))`

=> `lim_(h->0) (8x + 8h - 8x)/(h*(sqrt(8x + 8h) + sqrt (8x)))`

=> `lim_(h->0) (8h)/(h*(sqrt(8x + 8h) + sqrt (8x)))`

=> `lim_(h->0) 8/((sqrt(8x + 8h) + sqrt (8x)))`

substitute h = 0

=> `8/(sqrt 8x + sqrt (8x))`

=> `4/(sqrt (8x))`

=> `sqrt 2/sqrt x`

The required derivative is `sqrt 2/sqrt x`

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