A) \[\frac{y}{2}\left[ \frac{1}{x-a}+\frac{1}{x-b}-\frac{1}{x-c}-\frac{1}{x-d} \right]\]
B) \[y\,\left[ \frac{1}{x-a}+\frac{1}{x-b}-\frac{1}{x-c}-\frac{1}{x-d} \right]\]
C) \[\frac{1}{2}\left[ \frac{1}{x-a}+\frac{1}{x-b}-\frac{1}{x-c}-\frac{1}{x-d} \right]\]
D) None of these
Correct Answer: A
Solution :
\[y=\sqrt{\left[ \frac{(x-a)(x-b)}{(x-c)(x-d)} \right]}\] Þ\[\log y=\frac{1}{2}[\log (x-a)+\log (x-b)-\log (x-c)-\log (x-d)]\] Differentiating w.r.t. x we get \[\frac{1}{y}\frac{dy}{dx}=\frac{1}{2}\left[ \frac{1}{(x-a)}+\frac{1}{(x-b)}-\frac{1}{(x-c)}-\frac{1}{(x-d)} \right]\] Thus \[\frac{dy}{dx}=\frac{y}{2}\left[ \frac{1}{(x-a)}+\frac{1}{(x-b)}-\frac{1}{(x-c)}-\frac{1}{(x-d)} \right]\].
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