745. Find the derivatives of the following functions:
1) $\left.y^{3}-3 y+2 a x=0 ; 2\right) x^{2}+3 x y+y^{2}+1=0$. Calculate $y^{\prime}$ at the point $(2 ;-1) ;$; 3) $\sin \varphi+r \varphi-5 r=0$. Calculate $\frac{d r}{d \psi}$; 4) $e^{y}+x y=e$. Calculate $y^{\prime}$ at the point $(0 ; 1)$.
Solution. 1) Differentiating both sides of the equality with respect to $x$, taking into account that $y$ is a function of $x$, we get

$$
3 y^{2} y^{\prime}-3 y^{\prime}+2 a=0
$$

from which

$$
y^{\prime}=\frac{2 a}{3\left(1+y^{2}\right)}
$$

2) Differentiating with respect to $x$, we get

$$
2 x+3 y+3 x y^{\prime}+2 y y^{\prime}=0
$$

from which

$$
y^{\prime}=-\frac{2 x+3 y}{3 x+2 y}
$$

Now substituting $x=2, y=-1$, we find

$$
y^{\prime}(2)=-\frac{2 \cdot 2+3(-1)}{3 \cdot 2+2(-1)}=-\frac{1}{4}
$$

3) Differentiating with respect to $\varphi$, we get

$$
\cos \varphi+\frac{d r}{d \varphi} \cdot \varphi+r-5 \frac{d r}{d \varphi}=0
$$

from which

$$
\frac{d r}{d \varphi}=\frac{r+\cos \varphi}{5-\varphi}
$$

4) Differentiating with respect to $x$, we find

$$
e^{y} y^{\prime}+y+x y^{\prime}=0
$$

from which $y^{\prime}=-\frac{y}{e^{y}+x}, a y^{\prime}(0)=-\frac{1}{e}$.

Find the derivatives of the following implicit functions: