How do you find the total differential of $z = x^2y^2$?

Conditions

• The function is $z = x^2y^2$.
• Standard rules of partial differentiation apply.

Reasoning, step by step

1. Differentiate z with respect to x, treating y as a constant, to get ∂z/∂$x = 2$xy^2.
2. Differentiate z with respect to y, treating x as a constant, to get ∂z/∂$y = 2x^2y$.
3. Substitute these partial derivatives into the formula dz = (∂z/∂x)dx + (∂z/∂y)dy.
4. Combine the terms to get the final expression $dz = 2$xy^$2 dx + 2x^2y dy$.

Example

The instructor calculates the partial derivative with respect to x, treating y as a constant, resulting in 2xy^2. Next, he calculates the partial derivative with respect to y, treating x as a constant, yielding $2x^2y$. Combining these parts, the final total differential is written as $dz = 2$xy^$2 dx + 2x^2y dy$.

Common misconceptions

• Incorrectly applying the product rule when taking partial derivatives, instead of treating the other variable as a constant.
• Forgetting to multiply the partial derivatives by dx and dy respectively.

Watch the explanation

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