Question

Find the first partial derivative of the function z=x²y²(x⁴+y⁴)

Answer 1

To find the first partial derivative of the function z=x²y²(x⁴+y⁴), we differentiate each term separately and sum them up using the product and chain rules.

Step-by-step explanation:

To find the first partial derivative of the function z=x²y²(x⁴+y⁴), we need to differentiate each term separately and sum them up.

1. For the term x²y², we can differentiate it using the product rule: d(x²y²) = 2xy²dx + 2x²ydy.
2. For the term (x⁴+y⁴), we can differentiate it using the chain rule: d(x⁴+y⁴) = 4x³dx + 4y³dy.
3. Now, we can substitute these derivatives into the function to find: dz = (2xy²dx + 2x²ydy) * (x⁴+y⁴) + (4x³dx + 4y³dy) * x²y².
4. Finally, we can simplify the expression to get the first partial derivative of z with respect to x: ∂z/∂x = 2x⁷y² + 4x⁵y⁴ + 4x⁵y² + 4xy⁶.