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Question Number 93725 by ckkim89 last updated on 14/May/20

Answered by Rio Michael last updated on 14/May/20

$$\mathrm{let}y=f\left(x\right)\Rightarrow y={\cos }^{-1}(-3x)$$$$\Rightarrow \cos y=-3x$$$$\Rightarrow -\sin yy{}^{\prime }=-3$$$$\Rightarrow y^{\prime }=\frac{3}{\sin y}$$$$\mathrm{but}\sin y=\sqrt{1-{\cos }^{2}y}$$$$\Rightarrow y^{\prime }=\frac{3}{\sqrt{1-9x^2}}$$$$\Rightarrow f{}^{\prime }\left(x\right)=\frac{3}{\sqrt{1-9x^2}}$$

Commented by ckkim89 last updated on 14/May/20

thanks!

Answered by hknkrc46 last updated on 15/May/20

$$f\left(x\right)=y={\cos }^{-1}(-3x)\Rightarrow y^{{}^{\prime }}=-\frac{{(-3x)}^{{}^{\prime }}}{\sqrt{1-{(-3x)}^2}}$$$$=\frac{3}{\sqrt{1-9x^2}}$$$$\frac{d}{dx}{\cos }^{-1}\left[h\left(x\right)\right]=-\frac{h^{{}^{\prime }}\left(x\right)}{\sqrt{1-h^2\left(x\right)}}$$

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