<s>[INST] Derivation and tangent problem set Problem 1
On the curve $y=\frac{1}{1+x^2}$ find a point in which tangent line is parallel to the horizontal axis.

My idea:

Let's find $y'$. 
$$y'=\frac{-2x}{(1+x^2)^2}$$
If we want a tangent line to be parallel to x-axis it must look like this $t...y=kx+l$ such that $k=0$.
Let's find for which $x$ we have that situation.
$$y'=\frac{-2x}{(1+x^2)^2}=0$$
$$x=0\to y=1$$
$$(x,y)=(0,1)$$
Problem 2
Determine the angle of the curve
$y^2 - 3x^2 + x + 1 = 0$ and 
$xy^2 - 1 = 0$
intersect in the first quadrant
My work:
$y^2 - 3x^2 + x + 1 =xy^2 - 1$
One solution for $x>0$ and $y>0$ is $x=y=1$.
Let's now find derivative of easch function at that point.
$$2xyy'=-y^2$$
$$y'=\frac{-y^2}{2xy}=\frac{-1}{2}$$
$$2yy'=6x-1$$
$$y'=\frac{6x-1}{2y}=\frac{5}{2}$$
And the angle is $\alpha =arctg|\frac{k_1-k_2}{1+k_1*k_2}|$ where $k_1=\frac{-1}{2}$ and $k_2=\frac{5}{2}$

Problem 3
Find the angle of intersection of the tangents to the curve
$3x^2 - 6x + 2y^2 - 3 = 0$
from the point $T( 1,3 )$?
My work:
First (1,3) isn't on the curve.
We will need that tangent equation:
$$y-y_0=y'(x_0)(x-x_0)$$
From the task we know that $x=1$ and $y=3$.
Let's try to find $x_0$ and $y_0$. 
Using implict differentation we will find $y'$.
$$6x-6+4yy'=0$$
$$3x-3+2yy'=0$$
$$2yy'=-3x+3$$
$$y'=\frac{-3x+3}{2y}$$
Plugin that in equation:
$$3-y_0=\frac{-3x_0+3}{2y_0}(1-x_0)$$
$$3-y_0=\frac{-3x_0+3}{2y_0}+\frac{-3x_0^2-3x_0}{2y_0}$$
$$3-y_0=\frac{-3x_0^2+3-6x_0}{2y_0}$$
$$6y_0-2y_0^2=-3x_0^2+3-6x_0$$
But I can't get something nice...Is my problem 1 and 2 ok and please help me with 3
 [/INST] The solutions to problems 1) and 2) are correct.
For problem 3) note that you are searching the straight lines of equation $(y-3)=m(x-1)$ that are tangent to the curve $3x^2-6x+2y^2-3=0$. This means that you want the values of $m$ such that the system:
$$
\begin{cases}
(y-3)=m(x-1)\\
3x^2-6x+2y^2-3=0
\end{cases}
$$
has one double solution, and this is done if the discriminant of the second degree equation that we obtain substituting from the first to the second equation is null. This gives an equation ( of second degree) in $m$ and solving this equation you find the slopes of the tangent lines.
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