Prove that `d/(dx){x/2sqrt(a^2-x^2)+(a^2)/2sin^(-1)x/a}=sqrt(a^2-x^2)`

Explanation

`\text { To prove } \frac{d}{d x}\left(\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}\right)=\sqrt{a^{2}-x^{2}} `


`\text { L.H.S } \frac{d}{x}\left(\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}\right)`


`=\frac{d}{d x}\left(\frac{x}{2} \sqrt{a^{2}-x^{2}}\right)+\frac{a^{2}}{2} \frac{d}{d x} \sin ^{-1} \frac{x}{a} `


`\frac{x}{2} \frac{d}{d x} \sqrt{a^{2}-x^{2}}+\sqrt{a^{2}-x^{2}} \frac{d}{d x} \frac{x}{2}+\frac{a^{2}}{2} \frac{1}{\sqrt{1-(x / a)^{2}}} \cdot \frac{1}{a} `


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