Use the definition of the derivative (first principles) to find f '(x) of the function f(x) = 1/sqrt(x-1).

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to find the derivative of f(x) = `1/sqrt(x - 1)` from the first principles.

For any function f(x), f'(x) = `lim_(h->0)(f(x+h) - f(x))/h`

Here f(x) = `1/sqrt(x - 1)`

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

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

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

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

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

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

canceling h in the numerator and denominator and substituting h = 0

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

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

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

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