If $x^{\frac{2}{5}}+y^{\frac{2}{5}}=a^{\frac{2}{5}}$,then $\frac{dy}{dx} = $

If $e^x+e^y=e^{x+y}$,then $\frac{dy}{dx}=$

If $3 f(\cos x) + 2 f(\sin x) = 5 x$,then $f^{\prime}(\cos x) + f^{\prime}(\sin x) =$

Find $\frac{dx}{dy}$ for the equation $\sin^{2} y + \cos(xy) = \pi$.

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 $3.$

For $x > 1$,if $(2x)^{2y} = 4e^{2x-2y}$,then $(1 + \log_e 2x)^2 \frac{dy}{dx}$ is equal to