Maharashtra Board Solutions for Chapter: Differentiation, Exercise 3: EXERCISE 1.3

If ${\log }_5\left(\frac{x^4+y^4}{x^4-y^4}\right)=2$, show that $\frac{dy}{dx}=-\frac{12x^3}{13y^3}$.

If $e^x+e^y=e^{x+y}$, show that $\frac{dy}{dx}=-e^{y-x}$

If ${\sin }^{-1}\left(\frac{x^5-y^5}{x^5+y^5}\right)=\frac{\pi }{6}$, show that $\frac{dy}{dx}=\frac{x^4}{3y^4}$.

If $x^y=e^{x-y}$, then show that $\frac{dy}{dx}=\frac{\log x}{{\left(1+\log x\right)}^2}$.

If $y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\dots \infty }}}$, then show that $\frac{dy}{dx}=\frac{\sin x}{1-2y}$.

If $y=\sqrt{\log x+\sqrt{\log x+\sqrt{\log x+\dots \infty }}}$ then show that $\frac{dy}{dx}=\frac{1}{x\left(2y-1\right)}$.

If $y=x^{x^{x^{....to \infty }}}$ ,prove that $\frac{dy}{dx}=\frac{y^2}{x\left(1-y \log x\right)}.$

If $e^y=y^x,$ show that $\frac{dy}{dx}=\frac{(\log y{)}^2}{\log y-1}$.