Erson,

This is an exercise with using the chain rule. The key is to realize that cos³x is another way to write (cosx)³. Hence you differentiate cos3x much the same way you would differentiate (x² - 7x + 9)³. So let's do it.

If y = (x² - 7x + 9)³ then to find y' you first notice that y = (something)³ and you know how to differentiate something raised to the third power because if

y = x³ then y' = 3x².

But my function, y = (x² - 7x + 9)³ isn't quite that simple, it's y = (something)³ and the chain rule tells me that

y' = 3(x² - 7x + 9)² × (the derivative of x² - 7x + 9).

But I know the derivative of x² - 7x + 9, its 2x - 7. Thus what I have is that if

y = (x² - 7x + 9)³

then

y' = 3(x² - 7x + 9)² × (the derivative of x² - 7x + 9) = 3(x² - 7x + 9)² × (2x - 7).

Now you try this with y = cos³x = (cosx)³.

Harley