If $y = \cos^{2} \frac{3x}{2} - \sin^{2} \frac{3x}{2}$,then $\frac{d^{2}y}{dx^{2}}$ is

If $\left(\frac{dy}{dx}\right) = \frac{1}{\left(\frac{dx}{dy}\right)}$ and $\frac{d^2x}{dy^2}\left(\frac{dy}{dx}\right)^3 + \frac{d^2y}{dx^2} = k$, then $e^{k f(x)} - k f(x) =$

If $ y = (\tan^{-1} x)^2 $,then $ (x^2 + 1)^2 y_2 + 2x(x^2 + 1) y_1 $ is equal to

The second derivative of the function $y = f(x)$ is $f''(x) = 6(x - 1)$. If the graph of the function passes through the point $(2, 1)$ and the equation of the tangent at that point is $y = 3x - 5$,find the equation of the function.

If $x = A \cos 4t + B \sin 4t$,then $\frac{d^2x}{dt^2} = $

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