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Question Number 202761 by Mastermind last updated on 02/Jan/24

Answered by shunmisaki007 last updated on 03/Jan/24

$$g\left(x\right)=f^{-1}\left(x\right)$$$$f\left(x\right)=g^{-1}\left(x\right)$$$$f^{\prime }\left(x\right)=\sin \left(x\right)$$$$f\left(x\right)=c-\cos \left(x\right)\mathrm{where}c\mathrm{is}\mathrm{constant}.$$$$f\left(g\left(x\right)\right)=c-\cos \left(g\left(x\right)\right)=g^{-1}\left(g\left(x\right)\right)$$$$\cos \left(g\left(x\right)\right)=c-x(\Rightarrow g\left(x\right)={\cos }^{-1}(c-x))$$$$-\sin \left(g\left(x\right)\right)g^{\prime }\left(x\right)=-1$$$$\therefore g^{\prime }\left(x\right)=\frac{1}{\sin \left(g\left(x\right)\right)}=\frac{1}{\sin \left({\cos }^{-1}(c-x)\right)}\mathrm{where}c\mathrm{is}\mathrm{constant}.★$$$$$$

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