If , prove that

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Solution

Given, y=| f( x ) g( x ) h( x ) l m n a b c |.

Simplify and differentiate both sides with respect to x,

y=( mc−nb )f( x )−( lc−na )g( x )+( lb−ma )h( x ) dy dx = d dx { ( mc−nb )f( x )−( lc−na )g( x )+( lb−ma )h( x ) } =( mc−nb ) f ′ ( x )−( lc−na ) g ′ ( x )+( lb−ma ) h ′ ( x )

Now,

dy dx =| f ′ ( x ) g ′ ( x ) h ′ ( x ) l m n a b c |

Hence, the given condition is proved.

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