let h, k, x be Real; :: thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = k / (x ^2) ) & x + (h / 2) <> 0 & x - (h / 2) <> 0 holds
(cD (f,h)) . x = (- (((2 * h) * k) * x)) / (((x ^2) - ((h / 2) ^2)) ^2)
let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = k / (x ^2) ) & x + (h / 2) <> 0 & x - (h / 2) <> 0 implies (cD (f,h)) . x = (- (((2 * h) * k) * x)) / (((x ^2) - ((h / 2) ^2)) ^2) )
assume that
A1: for x being Real holds f . x = k / (x ^2) and
A2: ( x + (h / 2) <> 0 & x - (h / 2) <> 0 ) ; :: thesis: (cD (f,h)) . x = (- (((2 * h) * k) * x)) / (((x ^2) - ((h / 2) ^2)) ^2)
(cD (f,h)) . x = (f . (x + (h / 2))) - (f . (x - (h / 2))) by DIFF_1:5
.= (k / ((x + (h / 2)) ^2)) - (f . (x - (h / 2))) by A1
.= (k / ((x + (h / 2)) ^2)) - (k / ((x - (h / 2)) ^2)) by A1
.= ((k * ((x - (h / 2)) ^2)) - (k * ((x + (h / 2)) ^2))) / (((x + (h / 2)) ^2) * ((x - (h / 2)) ^2)) by A2, XCMPLX_1:130
.= (- (((2 * h) * k) * x)) / (((x ^2) - ((h / 2) ^2)) ^2) ;
hence (cD (f,h)) . x = (- (((2 * h) * k) * x)) / (((x ^2) - ((h / 2) ^2)) ^2) ; :: thesis: verum