Let f(x, y) = cos2 (x − y). Compute fx, fy, and fxx.

Question

Let f(x, y) = cos²(x − y). Compute fx, fy, and fxx.

Answer 1

To solve for partial derivatives we will treat the variable we are not using as a constant.

When solving for fx of f(x, y) = cos²(x − y), y will be treated as a constant, and using derivatives of trigonometric functions, we know that the derivative of cos²(x)=-sin(2x). We can also see is as (cos(x))² which means that the derivative can be achieved as 2cos(x) * (cos(x))' = 2cos(x)⋅(−sin(x))=−2cos(x)⋅sin(x)= −sin(2x) This leads to fx of f(x, y) = cos²(x − y)=-sin(2(x-y)).

fy of f(x, y) = cos²(x − y) has a slight change as the y in cos²(x − y) is negative. This leads to the partial derivative fy of (x-y) = -1. This means that a -1 is multiplied with the final answer to give us fy of f(x, y) = cos²(x − y)=sin(2(x-y)).

fxx of f(x, y) = cos²(x − y) is the same as fx of f(x, y)=-sin(2(x-y)), so we will start from there. The derivative of sin(x)= x'cos(x), so we will use this. Remember that y is still being seen as a constant for the partial derivative, which leads to fxx of f(x, y) = cos²(x − y) = fx of f(x, y)=-sin(2(x-y)) = -2cos(2(x-y)).