Problem Solution

Z = x² + y² represents a multivariable function. The partial derivative ^∂Z/^∂x treats y as constant, so differentiation yields ^∂Z/^∂x = 2x. Substituting x = 1 gives ^∂Z/^∂x = 2(1) = 2.

Step-by-Step Derivation

Differentiate term-by-term: the derivative of x² with respect to x is 2x, while y² contributes 0 as a constant. Evaluate at the point (1, 0): 2(1) + 0 = 2. No options are provided in the query, but common multiple-choice traps include total derivatives (2x + 2y = 2), second derivatives (2), or evaluation errors like 2(0) = 0 from confusing variables.

Introduction to Partial Derivative Z = x² + y²

In ^∂Z/^∂x Z = x² + y² X=1 Y=0 problems, students often seek the exact integer value for competitive exams like IIT JAM. This partial derivative calculation treats y as constant, making Z = x² + y² a simple quadratic in x. At X=1 and Y=0, the result is always the integer 2, crucial for multivariable calculus mastery.

Detailed Computation of ^∂Z/^∂x

Start with Z = x² + y². Compute ^∂Z/^∂x = 2x (y² vanishes). Plug in X=1: ^∂Z/^∂x = 2(1) = 2. Verify ^∂Z/^∂y = 2y = 0 at Y=0, confirming selective differentiation.

• First term: d(x²)/dx = 2x
• Second term: d(y²)/dx = 0 (constant)
• Evaluation: 2(1) + 0 = 2 (integer)

Common Errors in Partial Derivatives

Avoid mistaking it for total dZ/dx = 2x + 2y (equals 2 here but wrong method). Or confuse with ^∂²Z/^∂x^∂x = 2. At Y=0, no division-by-zero issues unlike √(x² + y²) cases.

Exam Relevance for IIT JAM

This ^∂Z/^∂x for Z = x² + y² at X=1 Y=0 tests basic partial differentiation, appearing in IIT JAM physics/chemistry sections. Practice yields quick integer answers like 2, building speed for tougher homogeneous functions.