If #f(x)= cos 4 x # and #g(x) = 2 x #, how do you differentiate #f(g(x)) # using the chain rule?
The chain rule is stated as:
Substituting the values on the property above:
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To differentiate ( f(g(x)) ) using the chain rule, first differentiate ( f(x) ) with respect to its inner function ( g(x) ), then multiply by the derivative of ( g(x) ).
( f(x) = \cos(4x) )
( g(x) = 2x )
Now, apply the chain rule:
- Find ( f'(x) ):
( f'(x) = -4\sin(4x) )
- Find ( g'(x) ):
( g'(x) = 2 )
- Substitute ( g(x) ) into ( f'(x) ) and multiply by ( g'(x) ):
( f'(g(x)) = -4\sin(4(2x)) = -4\sin(8x) )
( g'(x) = 2 )
So, ( (f \circ g)'(x) = -8\sin(8x) ).
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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.
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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