How can I find the first partial derivatives of the function f(x, y) = x²y?

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Finding the First Partial Derivatives of f(x, y) = x²y

To determine the first partial derivatives of the function f(x, y) = x²y, we will calculate the partial derivatives with respect to both variables, x and y.

1. Partial Derivative with respect to x

To find the partial derivative of f with respect to x, denoted as ∂f/∂x, we treat y as a constant. The function can be rewritten for clarity as:

f(x, y) = x² * y

Using the power rule of differentiation:

∂f/∂x = 2xy

So, the first partial derivative of the function with respect to x is:

∂f/∂x = 2xy

2. Partial Derivative with respect to y

Next, we find the partial derivative of f with respect to y, denoted as ∂f/∂y. Here, we treat x as a constant:

f(x, y) = x² * y

Using the basic rule that the derivative of y with respect to y is 1, we have:

∂f/∂y = x²

Thus, the first partial derivative of the function with respect to y is:

∂f/∂y = x²

Conclusion

In summary, the first partial derivatives of the function f(x, y) = x²y are:

  • ∂f/∂x = 2xy
  • ∂f/∂y = x²

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