Question

$\mathrm{If}f,g\mathrm{and}h\mathrm{are}\mathrm{differentiable}\mathrm{functions}\mathrm{of}x\mathrm{and}△\left(x\right)=\left(\begin{array}{ccc}f& g& h\\ \left(xf\right)\text{'}& \left(xg\right)\text{'}& \left(xh\right)\text{'}\\ \left(x^{2}f\right)\text{'}\text{'}& \left(x^{2}g\right)\text{'}\text{'}& \left(x^{2}h\right)\text{'}\text{'}\end{array}\right|,\mathrm{then}△\text{'}\left(x\right)\mathrm{is}$

• A. $\left(\begin{array}{ccc}f& g& h\\ f\text{'}& g\text{'}& h\text{'}\\ (x^{3}f\text{'}\text{'})\text{'}& (x^{3}h\text{'}\text{'})\text{'}& (x^{3}h\text{'}\text{'})\text{'}\end{array}\right|$
• B. $\left(\begin{array}{ccc}f& g& h\\ f\text{'}& g\text{'}& h\text{'}\\ (x^{3}f\text{'}\text{'})& (x^{3}h\text{'}\text{'})& (x^{3}h\text{'}\text{'})\end{array}\right|$
• C. $\left(\begin{array}{ccc}f& g& h\\ f\text{'}& g\text{'}& h\text{'}\\ (x^{3}f\text{'}\text{'})\text{'}\text{'}& (x^{3}h\text{'}\text{'})\text{'}\text{'}& (x^{3}h\text{'}\text{'})\text{'}\text{'}\end{array}\right|$
• D. $0$

Answer 1

The correct option is A

$\left(\begin{array}{ccc}f& g& h\\ f\text{'}& g\text{'}& h\text{'}\\ (x^{3}f\text{'}\text{'})\text{'}& (x^{3}h\text{'}\text{'})\text{'}& (x^{3}h\text{'}\text{'})\text{'}\end{array}\right|$

$\left(\begin{array}{ccc}f& g& h\\ xf\text{'}+f& xg\text{'}+g& xh\text{'}+h\\ 4xf\text{'}+2f+x^{2}f\text{'}\text{'}& 4xg\text{'}+2g+x^{2}g\text{'}\text{'}& 4xh\text{'}+2h+x^{2}h\text{'}\text{'}\end{array}\right|\mathrm{Operating}{R}_2\to R_1-R_1;R_3\to R_3-4R_2+2R_1\mathrm{and}\mathrm{shifting}x\mathrm{of}{R}_2\mathrm{to}{R}_3△\left(x\right)=\left(\begin{array}{ccc}f& g& h\\ f\text{'}& g\text{'}& h\text{'}\\ x^{3}f\text{'}\text{'}& x^{3}g\text{'}\text{'}& x^{3}h\text{'}\text{'}\end{array}\right|\Rightarrow △\text{'}\left(x\right)=0+0+\left(\begin{array}{ccc}f& g& h\\ f\text{'}& g\text{'}& h\text{'}\\ (x^{3}f\text{'}\text{'})\text{'}& (x^{3}g\text{'}\text{'})\text{'}& (x^{3}h\text{'}\text{'})\text{'}\end{array}\right|$