Tangent line to a parabola

Find the tangent line to the parabola $y = x^2$ at the point $(1, 1)$. Differentiating $y = x^2$ gives $y' = 2x$, so the slope at $x = 1$ is $2$. The tangent line is $y - 1 = 2(x - 1)$, that is $y = 2x - 1$.

That it is tangent is confirmed by $x^2 = 2x - 1$, i.e. $x^2 - 2x + 1 = (x - 1)^2 = 0$, whose double root $x = 1$ means the line meets the parabola at exactly one point. The large dot marks the point of tangency $(1, 1)$.