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Question Number 28305 by NECx last updated on 23/Jan/18

$$ify={\sin }^{-1}{x}^2+{\cos }^{-1}{x}^2$$$$finddy/dx$$

Commented by abdo imad last updated on 23/Jan/18

$$y\left(x\right)=arcsin\left(x^2\right)+arcos\left(x^2\right)$$$$y^{{}^{\prime }}\left(x\right)=\frac{2x}{\sqrt{1-x^4}}+\frac{-2x}{\sqrt{1-x^4}}=0\Rightarrow y\left(x\right)=\lambda \mathrm{\forall }x\in R$$$$y\left(0\right)=\frac{\pi }{2}\Rightarrow y\left(x\right)=\frac{\pi }{2}.$$

Commented by abdo imad last updated on 23/Jan/18

$$ifyoumeany\left(x\right)={\left(arcsinx\right)}^2+{\left(arcosx\right)}^2$$$$y^{{}^{\prime }}\left(x\right)=2\frac{arcsinx}{\sqrt{1-x^2}}-2\frac{arcosx}{\sqrt{1-x^2}}=\frac{2}{\sqrt{1-x^2}}(arcsinx-arcosx)$$$$=\frac{-2}{\sqrt{1-x^2}}(arcos(-x)+arcsin(-x))=\frac{-2}{\sqrt{1-x^2}}\frac{\xi \pi }{2}$$$$=\frac{-\pi \xi }{\sqrt{1-x^2}}with{\xi }^2=1and-1<x<1.$$

Commented by NECx last updated on 24/Jan/18

$$thankyousomuch.$$

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