Lecture 3 1d Kinematics

1. General Properties of the Derivative

1.1 Linearity of Derivatives

• Given functions f(t) and g(t) with constants A and B:

  • Expression:[ \frac{d}{dt}(Af(t) + Bg(t)) = A\frac{df}{dt} + B\frac{dg}{dt} ]

  • Proof:

    • Change in f and g:

      • [ \Delta f = f(t + \Delta t) - f(t) ]

      • [ \Delta g = g(t + \Delta t) - g(t) ]

    • Total change in expression:

      • [ \Delta (Af + Bg) = A \Delta f + B \Delta g ]

    • Taking the limit as ( \Delta t \to 0 ):

      • [ \frac{d}{dt}(Af + Bg) = \lim_{\Delta t \to 0} \frac{A\Delta f + B\Delta g}{\Delta t} = A\frac{df}{dt} + B\frac{dg}{dt} ]

2. Leibniz Rule for Products of Functions

2.1 Formula

• For two functions f and g:

  • Expression:[ \frac{d}{dt}(f \cdot g) = \left(\frac{df}{dt}\right) \cdot g + f \cdot \left(\frac{dg}{dt}\right) ]

2.2 Proof

• Define ( h(t) = f(t)g(t) ):

  • Change in h:

    • [ \Delta h = h(t + \Delta t) - h(t) = f(t + \Delta t)g(t + \Delta t) - f(t)