If $\sqrt{y + x} + \sqrt{y - x} = c$, then $\frac{dy}{dx} = f(x) - \sqrt{[f(x)]^2 - 1}$. Find $f(x)$.

If $xy = \tan^{-1}(xy) + \cot^{-1}(xy)$,then $\left(\frac{dy}{dx}\right)_{(4,2)} = ?$ (where $x, y \in \mathbb{R}$)

If $y=\sqrt{\cosh x+\sqrt{\cosh x+\dots}}$,then $\frac{d y}{d x}=$

Let $y$ be an implicit function of $x$ defined by ${x^{2x}} - 2{x^x}\cot y - 1 = 0$. Then $y'(1)$ equals

Consider the functions defined implicitly by the equation $y^3-3y+x=0$ on various intervals in the real line. If $x \in(-\infty,-2) \cup(2, \infty)$,the equation implicitly defines a unique real valued differentiable function $y=f(x)$. If $x \in(-2,2)$,the equation implicitly defines a unique real valued differentiable function $y=g(x)$ satisfying $g(0)=0$.
$1.$ If $f(-10 \sqrt{2})=2 \sqrt{2}$,then $f^{\prime \prime}(-10 \sqrt{2})=$
$(A)$ $\frac{4 \sqrt{2}}{7^3 3^2}$ $(B)$ $-\frac{4 \sqrt{2}}{7^3 3^2}$ $(C)$ $\frac{4 \sqrt{2}}{7^3 3}$ $(D)$ $-\frac{4 \sqrt{2}}{7^3 3}$
$2.$ The area of the region bounded by the curves $y=f(x)$,the $x$-axis,and the lines $x=a$ and $x=b$,where $-\infty < a < b < -2$,is
$(A)$ $\int_a^b \frac{x}{3\left((f(x))^2-1\right)} dx+bf(b)-af(a)$
$(B)$ $-\int_a^b \frac{x}{3\left((f(x))^2-1\right)} dx+bf(b)-af(a)$
$(C)$ $\int_a^b \frac{x}{3\left((f(x))^2-1\right)} dx-bf(b)+af(a)$
$(D)$ $-\int_a^b \frac{x}{3\left((f(x))^2-1\right)} dx-bf(b)+af(a)$
$3.$ $\int_{-1}^1 g^{\prime}(x) dx=$
$(A)$ $2g(-1)$ $(B)$ $0$ $(C)$ $-2g(1)$ $(D)$ $2g(1)$
Give the answer for questions $1, 2$ and $3.$

If $x^k + y^k = a^k$ $(a, k > 0)$ and $\frac{dy}{dx} + (\frac{y}{x})^{\frac{1}{3}} = 0$,then $k$ has the value