Differentiate the function f(x) = 2x^3 + (cos(x))^2 + e^x

When differentiating a function that is the sum of three different parts we can differentiate each part separately:

a) 2x³ is easy to differentiate. We remember the rule d/dx[ax^b] = abx^b-1. So

2x³ --> 6x²

b) (cos(x))² is a bit harder. We can use the chain rule, as we have a function raised to a power. The chain rule is:

d/dx[(g(x))ⁿ] = n(g(x))^n-1 * d/dx[g(x)]

Also we need to remember that cos(x) differentiates to -sin(x)

So we have that

(cos(x))² --> -2cos(x)sin(x).

c) e^x is the easiest of the lot: it doesnt change when differentiated. 

e^x --> e^x

Therefore the final answer is:

d/dx[f(x)] = 6x² - 2cos(x)sin(x) + e^x