# How do you find the derivative of #f(x)=sin^2(cos^2(x))#?

We have many instances of the chain rule here. The trick is to think of a function like an onion--the function contains other functions.

The first function, the most pressing one, is the squared function.

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To find the derivative of ( f(x) = \sin^2(\cos^2(x)) ), you can use the chain rule along with the derivatives of the trigonometric functions.

- Start by differentiating the outer function with respect to the inner function.
- Then, multiply by the derivative of the inner function with respect to ( x ).

Here are the steps:

- Let ( u = \cos^2(x) ).
- Find ( \frac{du}{dx} ).
- Now, differentiate ( \sin^2(u) ) with respect to ( u ), then multiply by ( \frac{du}{dx} ).

Derivative of ( \sin^2(u) ) with respect to ( u ): [ \frac{d}{du}(\sin^2(u)) = 2\sin(u)\cos(u) ]

Using the chain rule: [ \frac{d}{dx}(\sin^2(\cos^2(x))) = 2\sin(\cos^2(x))\cos(\cos^2(x)) \cdot (-\sin(x)\cos(x)) ]

Simplify the expression: [ \frac{d}{dx}(\sin^2(\cos^2(x))) = -2\sin(x)\cos(x)\sin(\cos^2(x))\cos(\cos^2(x)) ]

Therefore, the derivative of ( f(x) = \sin^2(\cos^2(x)) ) is ( -2\sin(x)\cos(x)\sin(\cos^2(x))\cos(\cos^2(x)) ).

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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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