If `f,g,a n d \ h` are differentiable functions of `x` and `d(x)=|[f,g,h],[(xf)',(xg)',(x h)'],[(x^(2)f)'',(x^2g)'',(x^2h)'']|` prove that `d^(prime)(x)=|[f,g,h],[f',g',h'],[(x^3f' ')',(x^3g' ')',(x^3h ' ')']|`

Options

A. 1
B. 2
C. 3
D. 4

Answer

C

Explanation

We have,
`Delta = |(f,g,h),((xf)',(xg)',(xh)'),((x^(2)f)'',(x^(2)g)'',(x^(2) h)'')|`
`rArr Delta = |(f,g,h),(xf' +f,xg' + g,xh' +h),(x^(2) f'' + 4xf' + 2f,x^(2) g'' 4x g' + 2g,x^(2) h'' + 4xh' + 2h)|` Applying `R_(2) rarr R_(2) - R_(1) and R_(3) rarr R_(3) - 2R`, we get
`rArr Delta = |(f,g,h),(xf',xg',xh'),(x^(2) f'' + 4xf',x^(2) g''+ 4xg',x^(2) h'' + 4xh'')|`
`rArr Delta = |(f,g,h),(xf',xg',xh'),(x^(2)f'',x^(2)g'',x^(2)h'')`| [ Applying `R_(3) rarr R_(3) - 4R_(2)`]
`rArr Delta = x |(f,g,h),(f',g',h'),(x^(2)f'',x^(2)g'',x^(2)h'')|` [Taking x common from `R_(2)`]
`rArr Delta = |(f,g,h),(f',g',h'),(x^(3)f',x^(3)g'',x^(3)h'')|` [Multiplying `R_(3)` by x]
`:. Delta' = |(f',g',h'),(f',g',h'),(x^(3)f'',x^(3)g'',x^(3)h'')| + |(f,g,h),(f'',g'',h''),(x^(3)f'',x^(3)g'',x^(3)h'')| + |(f,g,h),(f',g',h'),((x^(3)f'')',(x^(3)g'')',(x^(3)h'')')|`
`rArr Delta' = 0 + 0 + |(f,g,h),(f',g',h'),((x^(3)f'')',(x^(3)g'')',(x^(3)h'')')|`
`rArr Delta' = |(f,g,h),(f',g',h'),((x^(3)f'')',(x^(3)g'')',(x^(3)h'')')|`
`:. n =3`