The Chain Rule, Tangent Lines, and Kinematic Motion

The Chain Rule: If y is a differentiable function of u, and u is a differentiable function of x, then: dy/dx = (dy/du) · (du/dx). In particular, for a function raised to a power: d/dx [f(x)]ⁿ = n [f(x)]^{n - 1} · f'(x).

Tangent and Normal Lines: At any regular point P(x1, y1) lying on the curve y = f(x):

A. Tangent Line Equation: Calculate slope m = f'(x1). The tangent line equation is: y - y1 = m(x - x1). If m = 0, the tangent is horizontal (y = y1). If the denominator of f'(x1) is zero, the tangent is vertical (x = x1).

B. Normal Line Equation: Because the normal line is perpendicular to the tangent, its slope is m_normal = -1 / m. The normal line equation is: y - y1 = (-1 / m)(x - x1).

Kinematic Physical Applications: If s(t) represents the position vector of a particle at time t: (1) Instantaneous Velocity v(t) = ds/dt = s'(t). (2) Instantaneous Acceleration a(t) = dv/dt = d²s/dt² = s''(t). When a particle changes its direction of motion, it comes to instantaneous rest: v(t) = 0.