Question

Differentiation of under root a square minus x square upon a square + x square

Answer 1

Step-by-step explanation:

in this figure you have completed answer...

Answer 2

Answer:

Concept:

differentiation formulas such as $\frac{d}{dx} (x^{n}) = nx^{n-1}$

and $\frac{d}{dx}(\frac{u}{v} )= \frac{v\frac{d}{dx}u- u \frac{d}{dx} v }{v^{2} }$

Given:

let y= $\sqrt[]{\frac{(a^{2} -x^{2}) }{(a^{2} +x^{2} )} }$

To Find:

Derivative of $\sqrt[]{\frac{(a^{2} -x^{2}) }{(a^{2} +x^{2} )} }$

Step-by-step explanation:

y= ${\frac{(a^{2} -x^{2}) }{(a^{2} +x^{2} )} }^{\frac{1}{2} }$

$\frac{dy}{dx} =\frac{d}{dx} \sqrt[]{\frac{(a^{2} -x^{2}) }{(a^{2} +x^{2} )} }$

⇒ $\frac{1}{2} {\frac{(a^{2} -x^{2}) }{(a^{2} +x^{2} )} }^{\frac{1}{2} -1}$× $\frac{d}{dx} {\frac{(a^{2} -x^{2}) }{(a^{2} +x^{2} )} }$

⇒$\frac{1}{2} {\frac{(a^{2} -x^{2}) }{(a^{2} +x^{2} )} }^{\frac{-1}{2} }$× $\frac{(a^{2} +x^{2} )\frac{d}{dx}(a^{2} -x^{2})-(a^{2} -x^{2})\frac{d}{dx} (a^{2} +x^{2}) }{(a^{2}+x^{2} )^{2} }$

⇒$\frac{1}{2} {\frac{(a^{2} -x^{2}) }{(a^{2} +x^{2} )} }^{\frac{-1}{2} }$× $\frac{-2x(a^{2} +x^{2} )-2(a^{2} -x^{2} )}{(a^{2} +x^{2} )^{2} }$

⇒$\frac{1}{2} {\frac{(a^{2} -x^{2}) }{(a^{2} +x^{2} )} }^{\frac{-1}{2} }$×$\frac{-4xa^{2} }{(a^{2} +x^{2} )^{2} }$

⇒$\frac{-2xa^{2} }{\sqrt{a^{2} -x^{2} }(a^{2}+x^{2} ) ^{\frac{3}{2} } }$

So,

$\frac{dy}{dx}= \frac{-2xa^{2} }{\sqrt{a^{2} -x^{2} }(a^{2}+x^{2} ) ^{\frac{3}{2} } }$