Limits Continuity And Differentiability Question 71
Question: If $ y=\frac{(a-x)\sqrt{a-x}-(b-x)\sqrt{x-b}}{\sqrt{a-x}+\sqrt{x-b}} $ , then $ \frac{dy}{dx} $ wherever it is defined is
Options:
A) $ \frac{x+(a+b)}{\sqrt{(a-x)(x-b)}} $
B) $ \frac{2x-a-b}{2\sqrt{a-x}\sqrt{x-b}} $
C) $ -\frac{(a+b)}{2\sqrt{(a-x)(x-b)}} $
D) $ \frac{2x+(a+b)}{2\sqrt{(a-x)(x-b)}} $
Show Answer
Answer:
Correct Answer: B
Solution:
$ y=\frac{{{(a-x)}^{3/2}}+{{(x-b)}^{3/2}}}{\sqrt{a-x}+\sqrt{x-b}} $
$ =\frac{(\sqrt{a-x}+\sqrt{x-b})(a-x-\sqrt{a-x}\sqrt{a-b}+x-b)}{\sqrt{a-x}+\sqrt{x-b}} $
$ =a-b-\sqrt{a-x}\sqrt{x-b} $
or $ \frac{dy}{dx}=\frac{1}{2\sqrt{a-x}}\sqrt{x-b}-\frac{1}{2\sqrt{x-b}}\sqrt{a-x} $
$ =\frac{2x-a-b}{2\sqrt{a-x}\sqrt{x-b}} $
BETA
