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

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Answer 1 · 2018-07-18T10:41:06

Kindly refer to 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#.