# What is the derivative of #f(x) = xcos^3(x^2)+sin(x)#?

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To find the derivative of ( f(x) = x\cos^3(x^2) + \sin(x) ), we'll use the sum rule and the chain rule.

The derivative of ( x\cos^3(x^2) ) with respect to ( x ) involves two parts:

- Differentiating ( x ) with respect to ( x ), which gives ( 1 ).
- Applying the chain rule to differentiate ( \cos^3(x^2) ). This involves differentiating ( \cos^3(u) ) with respect to ( u ), where ( u = x^2 ), and then multiplying by the derivative of ( x^2 ) with respect to ( x ).

So, the derivative of ( x\cos^3(x^2) ) with respect to ( x ) is ( 1 \cdot \cos^3(x^2) - 3x^2\sin(x^2)\cos^2(x^2) ).

The derivative of ( \sin(x) ) with respect to ( x ) is ( \cos(x) ).

Putting it all together, the derivative of ( f(x) ) is:

[ f'(x) = \cos^3(x^2) - 3x^2\sin(x^2)\cos^2(x^2) + \cos(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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