What are the first partial derivatives of the function z = x sin(xy)?

by

First Partial Derivatives of the Function z = x sin(xy)

To find the first partial derivatives of the function z = x sin(xy), we will differentiate with respect to each variable while treating the other variable as a constant.

1. Partial Derivative with respect to x

We denote the partial derivative of z with respect to x as ∂z/∂x. Using the product rule, since z = x sin(xy) is a product of x and sin(xy), we differentiate:

    ∂z/∂x = sin(xy) + x ∂(sin(xy))/∂x

Next, we need to find the partial derivative of sin(xy) with respect to x using the chain rule:

    ∂(sin(xy))/∂x = cos(xy) * (∂>(xy)/∂x) = cos(xy) * y

Combining these results gives:

    ∂z/∂x = sin(xy) + xy cos(xy)

2. Partial Derivative with respect to y

Now, we compute the partial derivative of z with respect to y, denoted as ∂z/∂y. Here, we treat x as a constant:

    ∂z/∂y = x ∂>(sin(xy))/∂y

Using the chain rule, we differentiate sin(xy) with respect to y:

    ∂>(sin(xy))/∂y = cos(xy) * (∂>(xy)/∂y) = cos(xy) * x

So, we have:

    ∂z/∂y = x^2 cos(xy)

Final Results

In summary, the first partial derivatives of the function z = x sin(xy) are:

  • ∂z/∂x = sin(xy) + xy cos(xy)
  • ∂z/∂y = x^2 cos(xy)

Leave a Comment