Question

Differentiate `y=f(x)=1/sqrt(1-x)` using the definition of derivative `(dy)/(dx)=lim_(h->0)(f(x+h)-f(x))/h`?

Answer 1

`(dy)/(dx)=1/(2(1-x)sqrt(1-x))`

Explanation:

We have `y=f(x)=1/sqrt(1-x)`

then `f(x+h)=1/sqrt(1-x-h)`

and `f(x+h)-f(x)=1/sqrt(1-x-h)-1/sqrt(1-x)`

= `(sqrt(1-x)-sqrt(1-x-h))/(sqrt((1-x)(1-x-h)))`

= `(sqrt(1-x)-sqrt(1-x-h))/(sqrt((1-x)(1-x-h)))xx(sqrt(1-x)+sqrt(1-x-h))/(sqrt(1-x)+sqrt(1-x-h))`

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

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

and `(dy)/(dx)=lim_(h->0)(f(x+h)-f(x))/h`

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

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