# How do you differentiate #f(x)=sin(4x^2) # using the chain rule?

using the chain rule gives :

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To differentiate ( f(x) = \sin(4x^2) ) using the chain rule, you need to follow these steps:

- Identify the outer function and the inner function.
- Compute the derivative of the outer function with respect to the inner function.
- Compute the derivative of the inner function with respect to ( x ).
- Multiply the results of steps 2 and 3 together.

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

So, first, differentiate the outer function ( \sin(u) ) with respect to the inner function ( u = 4x^2 ), which gives ( \cos(4x^2) ).

Next, differentiate the inner function ( u = 4x^2 ) with respect to ( x ), which gives ( 8x ).

Finally, multiply the results together: ( \frac{d}{dx}(\sin(4x^2)) = \cos(4x^2) \cdot 8x = 8x \cos(4x^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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