How to find the derivative of f(x) = √x − x from first principles.?

1 Answer
Jul 18, 2018

Kindly refer to Explanation.

Explanation:

Let, f(x)=sqrtx-x.

Recall that, f'(x)=lim_(t to x){f(t)-f(x)}/(t-x)...............(ast).

With f(x)=sqrtx-x," we have, "f(t)=sqrtt-t.

:. {(f(t)-f(x)}=(sqrtt-t)-(sqrtx-x),

:." for "t!=x, {f(t)-f(x)}/(t-x)={(sqrtt-sqrtx)-(t-x)}/(t-x),

=(sqrtt-sqrtx)/(t-x)-(t-x)/(t-x),

={(sqrtt-sqrtx)}/{(sqrtt)^2-(sqrtx)^2}-1,

={(sqrtt-sqrtx)}/{(sqrtt-sqrtx)(sqrtt+sqrtx)}-1,

rArr {f(t)-f(x)}/(t-x)=1/(sqrtt+sqrtx)-1, (t!=x).

"Hence, from "(ast), f'(x)=lim_(t to x){1/(sqrtt+sqrtx)-1},

=1/(sqrtx+sqrtx)-1.

rArr f'(x)=1/(2sqrtx)-1.