If $y = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots \infty$,then $\frac{dy}{dx} = $

$f(x)$ and $g(x)$ are differentiable functions such that $\frac{f(x)}{g(x)} = c$,where $c$ is a non-zero constant. If $\frac{f^{\prime}(x)}{g^{\prime}(x)} = \alpha(x)$ and $\left(\frac{f(x)}{g(x)}\right)^{\prime} = \beta(x)$,then $\frac{\alpha(x) - \beta(x)}{\alpha(x) + \beta(x)} = $

The derivative of $f(x) = |x^2 - x|$ at $x = 2$ is

If $f(x) = 2x^2 + 3x - 5$,then the value of $f'(0) + 3f'(-1)$ is equal to

Let $f: R \rightarrow R$ be a differentiable function such that $|f(x) - f(y)| \leq 2|x - y|^{\frac{3}{2}}$ for all $x, y \in R$. If $f(0) = 1$,then $\int_0^1 f^2(x) dx = $

Find the derivative: $\frac{d}{dx} \sqrt{x \sin x}$