Applications Of Derivatives – Tangents and Normal — Formula Sheet

Slope of tangent = \( \frac{dy}{dx} \)

Slope of normal = \( -\frac{1}{\frac{dy}{dx}} = -\frac{dx}{dy} \)

Tangent: \( y - y_1 = \frac{dy}{dx}(x - x_1) \)

Normal: \( y - y_1 = -\frac{1}{\frac{dy}{dx}}(x - x_1) \)

Length of subtangent = \( \left| \frac{y}{y'} \right| \)

Length of subnormal = \( \left| y y' \right| \)

Rolle's Theorem: If \( f(a) = f(b) \), then \( f'(c) = 0 \) for some \( c \in (a,b) \)

Lagrange's MVT: \( f'(c) = \frac{f(b) - f(a)}{b - a} \)

The derivative \( \frac{dy}{dx} \) at a point \( (x_1, y_1) \) gives the slope of the tangent to the curve \( y = f(x) \) at that point.

The equation of a line with slope \( m \) passing through \( (x_1, y_1) \) is \( y - y_1 = m(x - x_1) \).

For normal

\( m = -\frac{1}{\frac{dy}{dx}} \) at \( (x_1, y_1) \).

Tangent parallel to x-axis ⇒ derivative = 0

Vertical tangent ⇒ x = x₁; horizontal normal ⇒ y = y₁

Subtangent = y / y′

Subnormal = y × y′

Normal length = y√(1 + (y′)²)