5 SCREEN 1
10 PRINT "Proof of Power Rule for Derivative"
15 PRINT "and Integral"
20 PRINT "Prove f'(x^n)=nx^n-1"
90 PRINT "f'(x)=limit h->0 f(x+h)-f(x)/h"
100 PRINT "by the binomial theorem"
110 PRINT "(x + h)^n=x^n + nhx^n-1 + nC2h^2x^n-2 +"
115 PRINT "... + nh^n-1x + h^n"
120 PRINT "so f'(x^n)=limit h->0 (x^n + nhx^n-1 +"
125 PRINT "nC2h^2x^n-2 +...+ nh^n-1x + h^n - x^n)/h"
130 PRINT "=nx^n-1"
140 PRINT "The integral is the reverse; (x^n+1)/n+1"