10 SCREEN 12
20 PRINT "Proof of Product Rule for Derivatives"
30 PRINT "Prove for h(x)=f(x)*g(x) h'(x)=[f(x)g(x)]'=f'(x)g(x)+f(x)g'(x)"
40 PRINT "h'(x)=lim "; CHR$(127); "x->0 h(x+"; CHR$(127); "x)-h(x)/"; CHR$(127); "x"
50 PRINT "=lim "; CHR$(127); "x->0 f(x+"; CHR$(127); "x)g(x+"; CHR$(127); "x)-f(x)g(x)/"; CHR$(127); "x"
60 PRINT "f'(x)=lim "; CHR$(127); "x->0 f(x+"; CHR$(127); "x)-f(x)/"; CHR$(127); "x and g'(x)=lim "; CHR$(127); "x->0 g(x+"; CHR$(127); "x)-g(x)/"; CHR$(127); "x"
70 PRINT "h'(x)=lim "; CHR$(127); "x->0 f(x+"; CHR$(127); "x)g(x+"; CHR$(127); "x)-f(x)g(x)/"; CHR$(127); "x"
80 PRINT "Add and subtract f(x)g(x+"; CHR$(127); "x) in the numerator and simplify"
90 PRINT "=lim "; CHR$(127); "x->0 f(x+"; CHR$(127); "x)g(x+"; CHR$(127); "x)+(f(x)g(x+"; CHR$(127); "x)-f(x)g(x+"; CHR$(127); "x))-f(x)g(x)/"; CHR$(127); "x"
100 PRINT "=lim "; CHR$(127); "x->0 [f(x+"; CHR$(127); "x)-f(x)/"; CHR$(127); "x]g(x+"; CHR$(127); "x)+[g(x+"; CHR$(127); "x)-g(x)/"; CHR$(127); "x]f(x)"
110 PRINT "=f'(x)g(x)+f(x)g'(x)"
120 PRINT "Example: h(x)=x^2cos(x)"
130 PRINT "h'(x)=(x^2)'cos(x)+(x^2)(cos(x))'"
140 PRINT "=2xcos(x)-(x^2)sin(x)"
150 SYSTEM