Find the derivative of the function. f(x) = x cos(6/x)

Answer 1

h(x) = x cos(6/x)

Apply Product Rule (f(x) * g(x) where x is f(x) and cos(6/x) is g(x)):

Reminder of Product Rule: (f(x)g(x))' = f(x) * g(x)' + g(x) * f(x)'

h(x)' = x * cos(6/x)' + cos(6/x) * x'

Solving [cos(6/x) * x']: x' = 1,

cos(6/x)*1 = cos(6/x)

Solving [x * cos(6/x)']:

Apply Chain Rule

Chain Rule Reminder: f(g(x))' = g(x)' * f'(g(x)) (where f(x) is cos(x) and g(x) is 6/x)

(6/x)' = -6/x^2 (power rule of -1)

cos(x)' = -sin(x)

x * cos(6/x) = 6/x^2(sin(6/x)) (negatives cancel out from -sin(x) and -6/x^2)

Final Solution: h(x)' = 6(sin(6/x))/x^2 + cos(6/x)

Answer 2

y = x cos (6/x)

= x (-sin(6/x) (-6/x^2) + cos(6/x)