# What is the derivative of? : #sin^2(x/2) \ cos^2(x/2)#

##### 2 Answers

#### Explanation:

Note that

So

Finding the derivative of this is a little simpler :)

Let's first find the derivative of

To this use the chain rule.

Let's say

So

However to calculate

Bringing it all together:

Using the same method for

So

Now combine these in the final step.

You could probably reduce this, but I think it's fine :)

#### Explanation:

Let:

# y = sin^2(x/2) \ cos^2(x/2) #

We could apply the product rule and chain rule but the expression can be significantly simplified using the sine double angle formula:

# sin 2A = 2sinAcosA iff sinAcosA = 1/2 sin 2A #

Thus we can write the initial expression as:

# y = (1/2sinx)^2 #

# \ \ = 1/4sin^2x #

So differentiating, using the chain rule, we get:

# dy/dx = (1/4)(2sinx)(cosx) #

# " " = (1/4)(2sinxcosx) #

Again using the sine double angle formula, we have:

# dy/dx = (1/4)sin(2x) #