The total differential of a function gives us insights into how the function changes with respect to its variables. In this case, we will find the total differential of the function z = x * cos(y) * cos(x).
The total differential, denoted as dz, can be calculated using the following formula:
dz = ∂z/∂x * dx + ∂z/∂y * dy + ∂z/∂x * dx
To find the total differential, we first need to compute the partial derivatives of z with respect to x, y, and the input x again. Let’s do this step-by-step:
Step 1: Compute ∂z/∂x
Using the product rule and chain rule, we differentiate z with respect to x:
∂z/∂x = cos(y) * cos(x) - x * cos(y) * sin(x)
Step 2: Compute ∂z/∂y
Next, we differentiate z with respect to y:
∂z/∂y = -x * sin(y) * cos(x)
Step 3: Combine into the total differential
Now that we have both partial derivatives, we can substitute them into our total differential formula:
dz = &left( cos(y) * cos(x) - x * cos(y) * sin(x) &right) dx + &left( -x * sin(y) * cos(x) &right) dy
So, the total differential of z = x * cos(y) * cos(x) is:
dz = &left( cos(y) * cos(x) - x * cos(y) * sin(x) &right) dx - x * sin(y) * cos(x) dy
In summary, the total differential of the function incorporates how changes in both x and y affect the value of z. Thus, utilizing this differential can help in various applications like approximating changes in values in engineering and science disciplines.