Q. Differentiate\[\frac{\sqrt{a^{2}+x^{2}}+\sqrt{a^{2}-x^{2}}}{\sqrt{a^{2}+x^{2}}-\sqrt{a^{2}-x^{2}}}\]
Differentiate \[ \frac{\sqrt{a^{2}+x^{2}}+\sqrt{a^{2}-x^{2}}}{\sqrt{a^{2}+x^{2}}-\sqrt{a^{2}-x^{2}}} \]
1 Answers
Let y = \(\frac{\sqrt{a^2+x^2}+\sqrt{a^2-x^2}}{\sqrt{a^2+x^2}-\sqrt{a^2-x^2}}\) \(\times\frac{\sqrt{a^2+x^2}+\sqrt{a^2+x^2}}{\sqrt{a^2+x^2}+\sqrt{a^2+x^2}}\)
⇒ y = \(\frac{(a^2+x^2)+(a^2-x^2)+2\sqrt{(a^2+x^2)(a^2-x^2)}}{(a^2+x^2)-(a^2-x^2)}\) (\(\because\) (a + b) (a - b) = a2 - b2 & (a + b)2 = a2 + 2ab + b2)
⇒ y = \(\frac{2a^2+2\sqrt{a^4-x^4}}{2x^2}\)
⇒ y = \(\frac{a^2+\sqrt{a^4-x^4}}{x^2}\)
\(\therefore\cfrac{dy}{dx}=\cfrac{x^2\left(\frac{1}{2\sqrt{a^4-x^4}}\times-4x^3\right)-2x(a^2+\sqrt{a^4-x^4})}{(x^2)^2}\)
\(=\cfrac{\frac{-2x^5}{\sqrt{a^4-x^4}}-2x(a^2+\sqrt{a^4-x^4})}{x^4}\)
\(=\frac{-2x^5-2xa^2\sqrt{a^4-x^4}-2xa^4+2x^5}{x^4\sqrt{a^4-x^4}}\)
\(=\frac{-2xa^2(\sqrt{a^4-x^4}+a^2)}{x^4\sqrt{a^4-x^4}}\)