Calculus

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2020 Q.28)** **1. Governing Concept & Definition**: - **Topic**: Engineering Mathematics & Partial Derivatives - For a multi-variable function $Z = f(X, Y)$, the partial derivative with respect to $X$ treats $Y$ as a constant: $\frac{\partial Z}{\partial X} = \frac{\partial}{\partial X}(X^2 + Y^2) = 2X$ **2. Detailed Step-by-Step Calculation**: - **Given**: $Z = X^2 + Y^2$ Point: $(X, Y) = (1, 0)$ - **Evaluate Partial Derivative**: $\left. \frac{\partial Z}{\partial X} \right|_{(1, 0)} = 2(1) = 2$ **3. Final Answer**: - **Calculated Value**: **2** - **Official GATE Accepted Range**: **2 to 2**