Answer-
The slope of the graph is 3e
Solution-
The given equation-
[tex]\Rightarrow x^4=\ln(xy)[/tex]
Using logarithm properties,
[tex]\Rightarrow x^4=\ln(x)+\ln(y)[/tex]
Taking derivatives of both sides,
[tex]\Rightarrow \dfrac{d}{dx}(x^4)=\dfrac{d}{dx}(\ln x+\ln y)[/tex]
[tex]\Rightarrow \dfrac{d}{dx}(x^4)=\dfrac{d}{dx}(\ln x)+\dfrac{d}{dx}(\ln y)[/tex]
Applying chain rule,
[tex]\Rightarrow \dfrac{d}{dx}(x^4)=\dfrac{d}{dx}(\ln x)+\dfrac{d}{dy}(\ln y)\dfrac{dy}{dx}[/tex]
[tex]\Rightarrow 4x^3=\dfrac{1}{x}+\dfrac{1}{y}\dfrac{dy}{dx}[/tex]
[tex]\Rightarrow \dfrac{1}{y}\dfrac{dy}{dx}=4x^3-\dfrac{1}{x}[/tex]
[tex]\Rightarrow \dfrac{dy}{dx}=y(4x^3-\dfrac{1}{x})[/tex]
Slope of [tex]x^4=\ln(xy)[/tex] at [tex](1,e)[/tex] is
[tex]=\dfrac{dy}{dx}_{at\ (1,e)}\\\\=e(4(1)^3-\dfrac{1}{1})\\\\=e(4-1)\\\\=3e[/tex]
