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Question Number 124007 by liberty last updated on 30/Nov/20

$$Givenafunctiony=f\left(x\right)where{f}^{-1}\left(\frac{x+5}{x-5}\right)=\frac{8}{x+5}$$$$Findslopeofthecurvey=f\left(x\right)atx=1.$$

Answered by john_santu last updated on 30/Nov/20

$$f^{-1}\left(\frac{x+5}{x-5}\right)=\frac{8}{x+5}\iff f\left(\frac{8}{x+5}\right)=\frac{x+5}{x-5}$$$$differentiatingbothsidegive$$$$-\frac{8}{{(x+5)}^2}f{}^{\prime }\left(\frac{8}{x+5}\right)=-\frac{10}{{(x-5)}^2}$$$$putx=3\Rightarrow -\frac{8}{64}f{}^{\prime }\left(1\right)=-\frac{10}{4}$$$$\Rightarrow f{}^{\prime }\left(1\right)=\frac{10}{4}\times \frac{8}{1}=20.$$

Answered by mathmax by abdo last updated on 30/Nov/20

$${\mathrm{f}}^{-1}\left(\frac{\mathrm{x}+5}{\mathrm{x}-5}\right)=\frac{8}{\mathrm{x}+5}\Rightarrow \mathrm{f}\left(\frac{8}{\mathrm{x}+5}\right)=\frac{\mathrm{x}+5}{\mathrm{x}-5}\mathrm{let}\frac{8}{\mathrm{x}+5}=\mathrm{t}\Rightarrow \mathrm{tx}+5\mathrm{t}=8\Rightarrow$$$$\mathrm{tx}=8-5\mathrm{t}\Rightarrow \mathrm{x}=\frac{8-5\mathrm{t}}{\mathrm{t}}\Rightarrow \mathrm{f}\left(\mathrm{t}\right)=\frac{\frac{8-5\mathrm{t}}{\mathrm{t}}+5}{\frac{8-5\mathrm{t}}{\mathrm{t}}-5}=\frac{8\mathrm{t}}{8-10\mathrm{t}}=\frac{4\mathrm{t}}{4-5\mathrm{t}}$$$$\mathrm{we}\mathrm{have}\mathrm{f}\left(\mathrm{t}\right)=\frac{4\mathrm{t}}{4-5\mathrm{t}}\Rightarrow {\mathrm{f}}^{{}^{\prime }}\left(\mathrm{t}\right)=\frac{4(4-5\mathrm{t})-4\mathrm{t}(-5)}{{(4-5\mathrm{t})}^2}=\frac{16-20\mathrm{t}+20\mathrm{t}}{{(5\mathrm{t}-4)}^2}$$$$\Rightarrow {\mathrm{f}}^{{}^{\prime }}\left(\mathrm{t}\right)=\frac{16}{{(5\mathrm{t}-4)}^2}\Rightarrow {\mathrm{f}}^{{}^{\prime }}\left(1\right)=16$$

Commented by liberty last updated on 30/Nov/20

$$notcorrect.$$$$f\left(t\right)=\frac{\frac{8-5t}{t}+5}{\frac{8-5t}{t}-5}=\frac{8-5t+5t}{8-5t-5t}=\frac{8}{8-10t}=\frac{4}{4-5t}$$$$sir=$$

Commented by Bird last updated on 30/Nov/20

$$sorryf\left(t\right)=\frac{8}{8-10t}=\frac{4}{4-5t}\Rightarrow$$$$f^{{}^{\prime }}\left(t\right)=4\times \frac{5}{{(4-5t)}^2}=\frac{20}{{(4-5t)}^2}\Rightarrow$$$$slope=f^{{}^{\prime }}\left(1\right)=20$$

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