1. ## derivative

what is the derivative of $\displaystyle cos^2 x$

2. Originally Posted by b00yeah05
what is the derivative of $\displaystyle cos^2 x$
From the chain rule, the answer is $\displaystyle -2 \cos x \sin x$, which is equivalent to $\displaystyle -\sin(2x)$.

3. hmmm...the part where i get confused is choosing which rule to use. by looking at that i thought it was the product rule...

How do you know when to use which rule?? i know the quotient rule because its usally something divided by something....

4. $\displaystyle cos^2 x$

is same as (cos x) multiply (cos x) so you use product rule

5. Originally Posted by b00yeah05
hmmm...the part where i get confused is choosing which rule to use. by looking at that i thought it was the product rule...

How do you know when to use which rule?? i know the quotient rule because its usally something divided by something....
You use the chain rule when you have a function of a function: y = f(g(x)).
You use the product rule when you have a product of two functions: y = f(x) times g(x).

Sometimes both rules can be used.

$\displaystyle y = \cos^2 x$ can be thought of as a function of a function: $\displaystyle y = (\cos x)^2$, that is, the square of $\displaystyle \cos x$. On the other hand .....

$\displaystyle y = \cos^2 x$ can be thought of as a product of two functions: $\displaystyle y = (\cos x) \times (\cos x)$.

So in this case it doesn't matter whether you use the chain rule or the product rule.

However, what would your plan be to differentiate $\displaystyle y = \cos^5 x$. Repeated use of the product rule? I hope NOT .....