Find $\frac{dy}{dx}$ for the equation $x^{2}+xy+y^{2}=100$.

If $y = x^{x^{x^{\dots\infty}}}$,then $\frac{dy}{dx} = $

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 $a(4+x^2)=x$ and $y-x^3=a^2$,then $\frac{dy}{dx}$ at $x=1$ is ...

If ${e^y} + xy = e$,the ordered pair $\left( {\frac{{dy}}{{dx}},\frac{{{d^2}y}}{{d{x^2}}}} \right)$ at $x = 0$ is equal to

If $3\sin (xy) + 4\cos (xy) = 5$,then $\frac{dy}{dx} = $