If $f(x)=1+nx+\frac{n(n-1)}{2}x^2+\frac{n(n-1)(n-2)}{6}x^3+\ldots+x^n$,then $f''(1)$ is equal to

Let $y = \frac{x^{2}}{(x+1)^{2}(x+2)}$. Then $\frac{d^{2} y}{dx^{2}}$ is

If $x=\frac{1-\sqrt{y}}{1+\sqrt{y}}$, then $(x+1) \frac{d^2 y}{d x^2}+\left(\frac{3 \sqrt{y}+1}{\sqrt{y}}\right) \frac{d y}{d x}$ equals

If $y^{1/4} + y^{-1/4} = 2x$,and $(x^2 - 1) \frac{d^2y}{dx^2} + \alpha x \frac{dy}{dx} + \beta y = 0$,then $|\alpha - \beta|$ is equal to ...... .

If $f(x)$ and $g(x)$ are twice differentiable functions on $(0,3)$ satisfying $f^{\prime \prime}(x)=g^{\prime \prime}(x)$, $f^{\prime}(1)=4$, $g^{\prime}(1)=6$, $f(2)=3$, and $g(2)=9$, then $f(1)-g(1)$ is

$A$ non-zero polynomial with real coefficients has the property that $f''(x) f'(x) = f(x)$. Then the value of $f'''(x)$ is