Calculus

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2023 Q.59)** **1. Governing Concept & Definition**: - **Topic**: Engineering Mathematics & Quotient Rule of Differentiation - For a rational function $f(x) = \frac{u(x)}{v(x)}$, the derivative is: $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$ **2. Detailed Step-by-Step Calculation**: - **Given Function**: $f(x) = \frac{\sin x + \cos x}{\sin x - \cos x}$ - $u(x) = \sin x + \cos x \implies u'(x) = \cos x - \sin x$ - $v(x) = \sin x - \cos x \implies v'(x) = \cos x + \sin x$ - **Evaluate components at $x = 0$**: - $\sin 0 = 0$, $\cos 0 = 1$ - $u(0) = 0 + 1 = 1$ - $u'(0) = 1 - 0 = 1$ - $v(0) = 0 - 1 = -1$ - $v'(0) = 1 + 0 = 1$ - **Substitution into Quotient Rule**: $f'(0) = \frac{(1)(-1) - (1)(1)}{(-1)^2} = \frac{-1 - 1}{1} = -2$ **3. Final Answer**: - **Calculated Value**: **-2** - **Official GATE Accepted Range**: **-2 to -2**