Find the angles at which the following curves intersect

Find the angles at which the following curves intersect the $Oy$ axis: 1° $y=\dfrac{x}{1+x^{2}}$; 2° $y=\dfrac{x}{\sqrt{3+x^{2}}}$.

Step 1: Part 1°

Both curves pass through the origin. For $y=\dfrac{x}{1+x^{2}}$: \[y' = \frac{1-x^{2}}{(1+x^{2})^{2}}, \qquad k = y'(0) = 1.\]

A line of slope $1$ has inclination $45°$; the angle it makes with the vertical $Oy$ axis (inclination $90°$) is

\[ \theta_{1} = 90°-45° = 45°. \]

Step 2: Part 2°

For $y=\dfrac{x}{\sqrt{3+x^{2}}}$, the quotient rule simplifies via $(3+x^2)-x^2=3$: \[y' = \frac{\sqrt{3+x^{2}}-x\cdot\dfrac{x}{\sqrt{3+x^{2}}}}{3+x^{2}} = \frac{3}{(3+x^{2})^{3/2}}, \qquad k = y'(0) = \frac{3}{3\sqrt3} = \frac{1}{\sqrt3}.\]

Inclination $\arctan\dfrac1{\sqrt3}=30°$; angle with $Oy$: $\theta_{2} = 90°-30° = 60°$.