Mathematics - Differentiation Question with Solution | TestHub

Consider the equation, $x{\log }_e\left({\log }_{e}x\right)-x^2+y^2=4$ Differentiate both sides w.r.t. $x$ ${\log }_e\left({\log }_{e}x\right)+x\cdot \frac{1}{x\cdot {\log }_{e}x}-2x+2y\frac{dy}{dx}=0$ ${\log }_e\left({\log }_{e}x\right)+\frac{1}{{\log }_{e}x}-2x+2y\frac{dy}{dx}=0$ When $x=e,y=\sqrt{4+e^2}$. Put these values in (1), $0+1-2e+2\sqrt{4+e^2}\frac{dy}{dx}=0$ $\frac{dy}{dx}=\frac{2e-1}{2\sqrt{4+e^2}}$