To find the first partial derivatives of the function z = x sin(xy), we need to take the derivatives with respect to each variable: x, y, and z.
Partial Derivative with respect to x (∂z/∂x)
We apply the product rule and the chain rule here:
- Let u = x and v = sin(xy).
Then, we have:
- ∂z/∂x = u’v + uv’
Where:
- u’ = 1 (derivative of x)
- v’ = cos(xy) * (y) (from chain rule)
Thus, substituting in:
- ∂z/∂x = 1 * sin(xy) + x * cos(xy) * y
So, the first partial derivative with respect to x is:
- ∂z/∂x = sin(xy) + xy cos(xy)
Partial Derivative with respect to y (∂z/∂y)
In this case, we regard x as a constant and differentiate with respect to y:
- z = x sin(xy)
Using the chain rule, we get:
- ∂z/∂y = x * cos(xy) * (x)
So, the first partial derivative with respect to y is:
- ∂z/∂y = x^2 cos(xy)
Final Result
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)