Let $f(x)$ be a differentiable function at $x=a$ with $f^{\prime}(a)=2$ and $f(a)=4$. Then $\lim _{x \rightarrow a} \frac{x f(a)-a f(x)}{x-a}$ equals ...... .

If $f:R \to R$ is a differentiable function and $f(1) = 4$,then the value of $\lim_{x \to 1} \int_4^{f(x)} \frac{2t}{x - 1} dt$ is:

If $y = \sec(\tan^{-1} x)$,then $\frac{dy}{dx}$ at $x = 1$ is

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. We say that $f$ has $PROPERTY \ 1$ if $\lim_{h \rightarrow 0} \frac{f(h)-f(0)}{\sqrt{|h|}}$ exists and is finite,and $PROPERTY \ 2$ if $\lim_{h \rightarrow 0} \frac{f(h)-f(0)}{h^2}$ exists and is finite. Then which of the following options is/are correct?
$(1) \ f(x)=x|x|$ has $PROPERTY \ 2$
$(2) \ f(x)=x^{2/3}$ has $PROPERTY \ 1$
$(3) \ f(x)=\sin x$ has $PROPERTY \ 2$
$(4) \ f(x)=|x|$ has $PROPERTY \ 1$

Differentiate $a^{x}$ with respect to $x$,where $a$ is a positive constant.

The derivative of $e^{ax} \cos bx$ with respect to $x$ is $re^{ax} \cos(bx + \alpha)$,where $\alpha = \tan^{-1}(\frac{b}{a})$. When $a > 0, b > 0$,the value of $r$ is: