If $x^2+y^2=t+\frac{1}{t}$ and $x^4+y^4=t^2+\frac{1}{t^2}$,then $\frac{dy}{dx}$ is equal to

If $x^{1/2} y^{1/3} = (x + y)^n$ and $x \frac{dy}{dx} - y = 0$, then $n =$

If $x^2 y^2 = \sin^{-1} x + \cos^{-1} x$,then $\frac{dy}{dx}$ at $x = 1$ and $y = 2$ is

Let $f : (0, \infty) \rightarrow (0, \infty)$ be a differentiable function such that $f(1) = e$ and $\lim_{t \rightarrow x} \frac{t^{2} f^{2}(x) - x^{2} f^{2}(t)}{t - x} = 0$. If $f(x) = 1$,then $x$ is equal to

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be two non-constant differentiable functions. If $f^{\prime}(x) = e^{(f(x)-g(x))} g^{\prime}(x)$ for all $x \in R$,and $f(1) = g(2) = 1$,then which of the following statement$(s)$ is (are) $TRUE$?

$If \frac{x}{x-y} = \log \left(\frac{a}{x-y}\right)$,then $\frac{dy}{dx} =$