Formula used :log mn = log m + log n log mn = n log mDifferentiation by part: \(\frac{d}{dx}(uv) = v\frac{du}{dx}+u\frac{dv}{dx}\)\(\frac{d}{dx}x^n=nx^{n-1}\)\(\frac{d}{dx}log\ x= \frac{1}{x}\) Calculation :Given thatxyyx = 1Taking the log on both sideslog( xyyx) = log 1Since, log mn = log m + log n ? log xy + log yx = log 1Since log mn = n log m? y log x + x log y = 0 (? log 1 = 0)Diff. with resepct to x? \(\frac{d}{dx}\)(y log x) + \(\frac{d}{dx}\) (x log y) = 0 Using differentiation by part? \(y\frac{d}{dx}log\ x+ log \ x \frac{dy}{dx}+ x\frac{d}{dx}log\ y+ log \ y \frac{dx}{dx}=0\)By using the formula (4) & (5)? \(\frac{y}{x}+ log \ x \frac{dy}{dx}+ \frac{x}{y}\frac{dy}{dx}+ log \ y=0\)? \(\frac{y}{x}+log\ y + \frac{dy}{dx}(\frac{x}{y}+log\ x) =0\)? \( \frac{dy}{dx}=-\frac{\frac{y}{x}+log\ y}{\frac{x}{y}+log\ x}\)Put x = y = 1 in above differential? \(\frac{dy}{dx}=-\frac{1 + log\ 1}{1 + log \ 1}= -1\) \(\frac{dy}{dx}\) at (1, 1) equal to -1.