For $x>1$,if $(2x)^{2y} = 4e^{2x-2y}$,then $(1+\log 2x)^2 \frac{dy}{dx}$ is equal to

Let $C$ be the curve $y^3 - 3xy + 2 = 0$. If $H$ and $V$ are the sets of points on the curve $C$ where the tangent to the curve is horizontal and vertical respectively,then

If $\log(\sqrt{1+x^2}-x) = y(\sqrt{1+x^2})$,then $(1+x^2) \frac{dy}{dx} + xy =$

The focal length of a mirror is given by $\frac{2}{f} = \frac{1}{v} - \frac{1}{u}$. In finding the values of $u$ and $v$, the errors are equal to $p$. Then, the relative error in $f$ is

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 ${x^{2/3}} + {y^{2/3}} = {a^{2/3}}$,then $\frac{dy}{dx} = $