If `f,g,a n dh` are differentiable functions of `xa n d(x)=|fgh(xf)'(xg)'(x h)'(x^(f2)f)' '(x^2g)' '(x^2h)' '|` prove that `^(prime)=|fgff'g' h '(x^3f' ')'(x^3g' ')'(x^3h ' ')'|`

Explanation

`(x f) =f+x f^{prime}
(x^{2} f)^{4} =[a 2 x f+x^{2} f^{prime}]^{prime}`


`=[2 f+2 x f^{prime}+2 x f^{prime}+x^{2} f^{prime prime}] `


`=[2 f+4 x f^{prime}+x^{2} f^{prime prime}]`


`Delta=[[f,g,h],[xf^(prime)+f , xg^(prime)+g , xh^(prime)+h] ,[2f+4f^(prime)x+x^2f^(prime prime) , 2g+4xg^(prime)+x^2g^(prime prime ) , 2h+4xh^(prime)+x^2h^(prime prime)]]`
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