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

If $f$ and $g$ are differentiable functions such that $g'(a) = 2$ and $g(a) = b$,and if $f \circ g$ is an identity function,then $f'(b)$ has the value equal to:

Find the derivative of the following function: $\frac{\cos x}{1+\sin x}$

$f(x)$ 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)} = $

If $f(x)=3 e^{x^2}$ then $f^{\prime}(x)-2 x f(x)+\frac{1}{3} f(0)-f^{\prime}(0)=$

Find the derivative of the following function (it is to be understood that $a, b, c,$ and $d$ are fixed non-zero constants): $\frac{a x+b}{c x+d}$