BCA Semester 2 – Mathematics II – Unit 2: Differentiation (6Hrs)

$\textbf{Given the parametric equations:}$ \[ x = a(t + \sin t) \] \[ y = a(1 - \cos t) \] To find \( \dfrac{dy}{dx} \), we use the formula: \[ \dfrac{dy}{dx} = \dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}

$\textbf{L'Hospital's Rule}$ If \[ \lim{x \to a} \frac{f(x)}{g(x)} \] is of the indeterminate form \( \dfrac{0}{0} \) or \( \dfrac{\infty}{\infty} \), then \[ \lim{x \to a} \frac{f(x)}{g(x)} \] \[

$\textbf{Step 1: Definition of derivative}$ The derivative of \( y \) with respect to \( x \) is given by: \[ \frac{dy}{dx} = \lim{h \to 0} \frac{f(x+h) - f(x)}{h} \] Here, \[ f(x) = e^{3x-1} \] ---

(a) Parametric differentiation Given \( x = t^2 - 1, \quad y = t^4 - 1 \) \[ \frac{dx}{dt} = 2t, \quad \frac{dy}{dt} = 4t^3 \] \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{4t^3}{2t}