Final answer:
The partial derivative fₓ of the given function f(x, y) can be found using the chain rule. The derivative of sin (x+2y) + x cos (x+2y) with respect to x is sin (x+2y) + x cos (x+2y). Therefore, fₓ = sin (x+2y) + x cos (x+2y).
Step-by-step explanation:
To find fₓ, we need to find the partial derivative of f(x, y) with respect to x. Since f(x, y) = x sin (x+2y) + cos (x+2y), we can use the chain rule to find fₓ. The chain rule states that if we have a function of the form f(g(x)), then the derivative of f(g(x)) with respect to x is f'(g(x)) * g'(x). Applying this to our function, f'(x) = sin (x+2y) + x cos (x+2y) * (1+0). Simplifying further, we get f'(x) = sin (x+2y) + x cos (x+2y). Therefore, fₓ = sin (x+2y) + x cos (x+2y).