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Q.
Differential coefficient of {xl+mm−n}1n−l⋅{xm+nn−l}1l−m⋅{xn+ll−m}1m−n with respect to x is:
NTA AbhyasNTA Abhyas 2022
Solution:
Let P = ({x(l+mm−n)})1n−l({x(m+nn−l)})1l−m({x(n+ll−m)})1m−n
As (ab)c=abc , we get, P={x((l+m)(m−n)(n−l))}{x((m+n)(n−l)(l−m))}{x((n+l)(l−m)(m−n))}
As (ab)(ac)=ab+c , we get, P=x{(l+m)(m−n)(n−l)+(m+n)(n−l)(l−m)+(n+l)(l−m)(m−n)}
Solving the exponent of x we get, (l+m)(m−n)(n−l)+(m+n)(n−l)(l−m)+(n+l)(l−m)(m−n) =(l+m)(l−m)+(m+n)(m−n)+(n+l)(n−l)(m−n)(n−l)(l−m) =(l2−m2)+(m2−n2)+(n2−l2)(m−n)(n−l)(l−m) =0
Hence, P=x0 . P=1 dPdx=ddx(1) P′(x)=0
Hence, the differential coefficient of P is 0 .