The Derivative of a Function f : RR to RR at a point x in RR, is
denoted by f'(x), and, is defined by,
lim_(t to x) (f(t)-f(x))/(t-x), if, the Limit exists.
We have, f(x)=4/sqrtx rArr f(t)=4/sqrtt.
Hence, (f(t)-f(x))/(t-x)=(4/sqrtt-4/sqrtx)/(t-x)={4(sqrtx-sqrtt)}/{sqrtt*sqrtx*(t-x)}.
:. lim_(t to x)(f(t)-f(x))/(t-x)=lim_(t to x){4(sqrtx-sqrtt)}/{sqrtt*sqrtx*(t-x)},
=lim{-4(sqrtt-sqrtx)}/{sqrtt*sqrtx*(t-x)}xx(sqrtt+sqrtx)/(sqrtt+sqrtx),
=lim(-4cancel((t-x)))/{sqrtt*sqrtx*cancel((t-x))*(sqrtt+sqrtx)},
=lim_(t to x)-4/{sqrtt*sqrtx*(sqrtt+sqrtx)},
=-4/{sqrtx*sqrtx*(sqrtx+sqrtx)},
=-4/(x*2sqrtx).
Thus, the Limit exists, and, therefore, for f(x)=4/sqrtx, we have,
f'(x)=-2*x^(-3/2); x >0..
Enjoy Maths.!