To solve this problem, let's follow these steps:
Step 1: Identify the curve and the point
The curve in question is given by the function:
f(x, y) = 6sin(x) + 5cos(y) - 2sin(x)cos(y) = 6π
The point at which we're wanting to find the slope is (6π, (3π)/2).
Step 2: Compute the derivative
We will take the derivative of the curve function with respect to x:
f'(x, y) = df/dx = -2cos(x)cos(y) + 6cos(x)
This derivative function tells us the slope of the tangent line to any point on the curve.
Step 3: Evaluate the derivative at the point
Now, we want to find the slope of the curve at a particular point, (6π, (3π)/2). Plug x = 6π and y = (3π)/2 into the derivative function:
f'(6π, (3π)/2) = -2cos(6π)cos((3π)/2) + 6cos(6π) = 6
So, the slope of the tangent line to the curve at the point (6π, (3π)/2) is 6.