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Prove by induction that {prod_(r=0)^(n)f_(r)(x)}'=sum_(i=1)^(n){f_1(x)f_2(x)....f_(i)'(x)....f_(n)(x)}, where dash denotes derivative with respect to x. |
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Answer» <P> Solution :Let `P(n):{prod_(r=0)^(n)f_(r)(x)}^(')=sum_(i=1)^(n){f_(1)(x)f_(2)(x).....f_(1)(x)....f_(n)(x)}`Step I For ` n =1` , LHS of EQ. (i) `={prod_(r=1)^(1)f_(r)F(x)}^(')={f_(1)(x)}^(')=f_(1)^(')(x)` RHS of Eq. (i) `=sum_(i=1)^(1){f_1(x)f_(2)(x)...f_(1)^(')(x)... f_(1)(x)}` which is true for `n=1`. Step II Assume it is true for `n=K` , then `P(k):{prod_(r=1)^(k)f_(r)(x)}^(')=sum_(i=1)^(k){f_1(x)f_2(x).....f_1(x)....f_k(x)}` Step III For `n=k+1`, LHS`={prod_(r=1)^((k+1))f_r(x)}={prod_(r=1)^(k)f_r(x).f_(k+1)(x)}^(')` `=prod_(r=1)^(k)f_r(x).f_(k+1)^(')(x)+f_(k+1)(x){prod_(r=1)^(k)f_r(x)}^'` `= prod_(r=1)^(k)f_r(x).f_(k+1)^(')(x)+f_(k+1)(x).sum_(i=1)^(k){f_1(x).f_2(x).....f_(k+1)^(')....f_(k)(x)}` `={f_1(x)f_2(x)....f_k(x)}f_(k+1)^(')(x)+f_(k+1)(x) sum_(i=1)^(k){f_1(x)f_2(x)....f_(i) '(x).....f_(k)(x)}` `= sum_(i=1)^(k+1){f_19x)f_2(x).....f_(i)'(x)....f_(k+1)(x)}=RHS` This shows that the result is true for `n=k+1`. Hence , by the principle of mathematical induction , the result is true for all `n in N`. |
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