Question

Differential coefficient of ${\left(x^{\frac{l+m}{m-n}}\right)}^{\frac{1}{n-l}}.{\left(x^{\frac{m+n}{n-l}}\right)}^{\frac{1}{l-m}}.{\left(x^{\frac{n+l}{l-m}}\right)}^{\frac{1}{m-n}}$ w.r.t. $x$ is:

• A. $1$
• B. $0$
• C. $-1$
• D. $x^{lmn}$

Answer 1

Answer: (B)

$0$

Let, $y={\left(x^{\frac{l+m}{m-n}}\right)}^{\frac{1}{n-l}}.{\left(x^{\frac{m+n}{n-l}}\right)}^{\frac{1}{l-m}}.{\left(x^{\frac{n+l}{l-m}}\right)}^{\frac{1}{m-n}}$

$=x^{(\frac{l+m}{(m-n)(n-l)}+\frac{m+n}{(n-l)(l-m)}+\frac{n+l}{(l-m)(m-n)})}$

$=x^{\frac{l^2-m^2+m^2-n^2+n^2-l^2}{(m-n)(n-l)(l-m)}}$

$=x^0=1$

$\therefore \frac{dy}{dx}=0$