To find the total differential of the function z = x * cos(y) * cos(x), we need to compute the partial derivatives of z with respect to each variable: x and y.
1. **Partial Derivative with respect to x**:
Using the product rule, we calculate:
∂z/∂x = cos(y) * cos(x) + x * cos(y) * (-sin(x))
So, ∂z/∂x = cos(y) * cos(x) - x * cos(y) * sin(x).
2. **Partial Derivative with respect to y**:
For y, we treat x as a constant:
∂z/∂y = x * (-sin(y)) * cos(x)
Thus, ∂z/∂y = -x * sin(y) * cos(x).
3. **Total Differential dz**:
The total differential dz can be expressed as:
dz = (∂z/∂x)dx + (∂z/∂y)dy
Substituting the partial derivatives we found:
dz = (cos(y) * cos(x) - x * cos(y) * sin(x))dx - (x * sin(y) * cos(x))dy
So, the total differential of the function z = x * cos(y) * cos(x) is:
dz = (cos(y) * cos(x) - x * cos(y) * sin(x))dx - (x * sin(y) * cos(x))dy