How do you find dy/dx: y= (8 radical x) + 10x - 32x + (x^2 cos2x) / x^2

Since differentation is linear, you need to calculate the derivative of each term. 8Sqrt(x) = 8(x)^(1/2) Recall the power rule and this becomes 8(1/2(x^-.5)) = 4/sqrt(x). 10x by the power rule is 10. -32x is the same rule, so we get -32

x^2cos(2x)/x^2 = cos(2x) and by the chain rule we get -sin(2x)*2

combining everything

4/sqrt(x) +10 -32 -2sin(2x)

8*x^(1/2) =>4x^(-1/2)
10x =>10
-32 =>0
x^2cos(2x)*x^-2 => x^2 and x^-2 cancel
cos(2x) => -2sin(2x) using the chain rule


4/x^(1/2)-2sin(2x)+10

THAT is the correct answer with your update that 32x is really 32 ***

Answering your second question: "also how do you find dy/dx: y=cos(11x^5 + 5x)"

use the chain rule:

y' = -sin(11x^5 + 5x)*(55x^4 + 5)

is that calc 2?