# If #f(x)= cos 4 x # and #g(x) = -3x #, how do you differentiate #f(g(x)) # using the chain rule?

According to chain rule

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To differentiate ( f(g(x)) ) using the chain rule, follow these steps:

- Identify the outer function ( f(x) ) and the inner function ( g(x) ).
- Calculate the derivative of the outer function ( f'(x) ).
- Calculate the derivative of the inner function ( g'(x) ).
- Substitute ( g(x) ) and ( g'(x) ) into ( f'(g(x)) ).
- Multiply ( f'(g(x)) ) by ( g'(x) ) to get the final result.

Given ( f(x) = \cos^4(x) ) and ( g(x) = -3x ):

- The outer function is ( f(x) = \cos^4(x) ) and the inner function is ( g(x) = -3x ).
- The derivative of the outer function is ( f'(x) = 4\cos^3(x) \cdot (-\sin(x)) ).
- The derivative of the inner function is ( g'(x) = -3 ).
- Substitute ( g(x) = -3x ) into ( f'(x) ) to get ( f'(-3x) = 4\cos^3(-3x) \cdot (-\sin(-3x)) ).
- Multiply ( f'(-3x) ) by ( g'(x) = -3 ) to get the final result:

[ f'(g(x)) \cdot g'(x) = 4\cos^3(-3x) \cdot (-\sin(-3x)) \cdot (-3) ]

Therefore, the derivative of ( f(g(x)) ) with respect to ( x ) is:

[ -12\cos^3(-3x) \cdot \sin(-3x) ]

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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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