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Question Number 119565 by bemath last updated on 25/Oct/20

Commented by bemath last updated on 25/Oct/20

$$gavekudos$$

Answered by TANMAY PANACEA last updated on 25/Oct/20

$$y={log}_7{\left(\frac{x+6}{x+6}\right)}^{ln7}$$$$y=ln7\times \frac{ln\left(\frac{x+6}{x-6}\right)}{ln7}[{log}_{a}b=\frac{logb}{loga})$$$$y=ln(x+6)-ln(x-6)$$$$\frac{dy}{dx}=\frac{1}{x+6}-\frac{1}{x-6}=\frac{x-6-x-6}{(x+6)(x-6)}$$$$\frac{dy}{dx}=\frac{-12}{(x+6)(x-6)}$$

Answered by Ar Brandon last updated on 25/Oct/20

$$\mathrm{y}={\log }_7\left[{\left(\frac{\mathrm{x}+6}{\mathrm{x}-6}\right)}^{\ln 7}\right]$$$$7^{\mathrm{y}}={\left(\frac{\mathrm{x}+6}{\mathrm{x}-6}\right)}^{\ln 7}={(1+\frac{12}{\mathrm{x}-6})}^{\ln 7}$$$$\Rightarrow 7^{\mathrm{y}}\ln 7\frac{\mathrm{dy}}{\mathrm{dx}}=-\ln 7\frac{12}{{(\mathrm{x}-6)}^2}{\left(\frac{\mathrm{x}+6}{\mathrm{x}-6}\right)}^{\ln 7-1}$$$$\Rightarrow 7^{\mathrm{y}}\frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{12}{(\mathrm{x}-6)(\mathrm{x}+6)}{\left(\frac{\mathrm{x}+6}{\mathrm{x}-6}\right)}^{\ln 7}$$$$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-12}{(\mathrm{x}-6)(\mathrm{x}+6)}$$

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