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

This could definitely be decreased, but in my opinion, it's okay.

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Let:

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

Thus we can write the initial expression as:

So differentiating, using the chain rule, we get:

Again using the sine double angle formula, we have:

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To find the derivative of sin²(x/2) * cos²(x/2), we can use the product rule. The product rule states that if we have two functions, f(x) and g(x), then the derivative of their product is f'(x) * g(x) + f(x) * g'(x), where f'(x) represents the derivative of f(x) and g'(x) represents the derivative of g(x).

Let's denote sin²(x/2) as f(x) and cos²(x/2) as g(x).

Now, we need to find the derivatives of f(x) and g(x) with respect to x:

f(x) = sin²(x/2) f'(x) = 2 * sin(x/2) * (1/2) * cos(x/2) = sin(x/2) * cos(x/2)

g(x) = cos²(x/2) g'(x) = 2 * cos(x/2) * (1/2) * (-sin(x/2)) = -sin(x/2) * cos(x/2)

Now, applying the product rule:

f'(x) * g(x) + f(x) * g'(x) = (sin(x/2) * cos(x/2)) * cos²(x/2) + sin²(x/2) * (-sin(x/2) * cos(x/2))

Simplifying further:

= sin(x/2) * cos(x/2) * cos²(x/2) - sin²(x/2) * sin(x/2) * cos(x/2)

= sin(x/2) * cos(x/2) * cos²(x/2) - sin³(x/2) * cos(x/2)

So, the derivative of sin²(x/2) * cos²(x/2) with respect to x is sin(x/2) * cos(x/2) * cos²(x/2) - sin³(x/2) * cos(x/2).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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