Tangent line at (x0, y0) on y = f(x): y - y0 = f'(x0)(x - x0)

Normal line at (x0, y0): y - y0 = -1/f'(x0) * (x - x0) [perpendicular to tangent]

Special cases:

• If f'(x0) = 0: tangent is horizontal (y = y0), normal is vertical (x = x0)
• If f'(x0) = infinity: tangent is vertical (x = x0), normal is horizontal (y = y0)

Angle between curves: Two curves y = f(x) and y = g(x) intersect at a point. The angle between them is the angle between their tangents: tan(theta) = |f'(x0) - g'(x0)|/(1 + f'(x0)*g'(x0)).

Orthogonal curves: Two curves are orthogonal if their tangents are perpendicular at the intersection: f'(x0) * g'(x0) = -1.

Length of tangent, normal, subtangent, subnormal:

• Subtangent = |y/f'(x)| (horizontal projection under tangent)
• Subnormal = |y*f'(x)| (horizontal projection under normal)
• Length of tangent = |y|*sqrt(1 + $\frac{1}{f'(x}$)^2)/1
• Length of normal = |y|*sqrt(1 + (f'(x))^2)