Question

The total differential \(df\) for the function \(f(x, y, z) = x^2 y^3 z^4\) is:

• A. \(df = (2x y^3 z^4) \, dx + (3x^2 y^2 z^4) \, dy + (4x^2 y^3 z^3) \, dz\)
• B. \(df = (2x y^3 z^4) \, dx + (3x^2 y^2 z^4) \, dy + (4x^2 y^3 z^3) \, dz\)
• C. \(df = (2x y^3 z^4) \, dx + (3x^2 y^2 z^4) \, dy + (4x^2 y^3 z^3) \, dz\)
• D. \(df = (2x y^3 z^4) \, dx + (3x^2 y^2 z^4) \, dy + (4x^2 y^3 z^3) \, dz\)

Hint:

Exam Tip:
For total differential:

• Use the formula \(df = f_x dx + f_y dy + f_z dz\).
• Apply the power rule for differentiation.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The total differential of a function \(f(x, y, z)\) is given by: \[ df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dz \]

Step 2: Key Formula or Approach:
Given \(f(x, y, z) = x^2 y^3 z^4\), compute the partial derivatives: \[ \frac{\partial f}{\partial x} = 2x y^3 z^4, \quad \frac{\partial f}{\partial y} = 3x^2 y^2 z^4, \quad \frac{\partial f}{\partial z} = 4x^2 y^3 z^3 \]

Step 3: Detailed Explanation:
So, the total differential is: \[ df = (2x y^3 z^4) dx + (3x^2 y^2 z^4) dy + (4x^2 y^3 z^3) dz \] This matches option (A).

Step 4: Final Answer:
Therefore, option (A) is correct.