If #f(x)= sin3x # and #g(x) = 2x^2 #, how do you differentiate #f(g(x)) # using the chain rule?
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To differentiate ( f(g(x)) ) using the chain rule, follow these steps:
 Differentiate the outer function ( f(x) ) with respect to its inner function ( g(x) ).
 Multiply by the derivative of the inner function ( g'(x) ).
Given ( f(x) = \sin(3x) ) and ( g(x) = 2x^2 ), we proceed as follows:

Compute the derivative of ( f(x) = \sin(3x) ): [ f'(x) = \frac{d}{dx}(\sin(3x)) = 3\cos(3x) ]

Compute the derivative of ( g(x) = 2x^2 ): [ g'(x) = \frac{d}{dx}(2x^2) = 4x ]

Apply the chain rule: [ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) ]

Substitute the derivatives we found: [ \frac{d}{dx}[f(g(x))] = 3\cos(3x) \cdot 4x ]
So, the derivative of ( f(g(x)) ) with respect to ( x ) is ( 12x\cos(3x) ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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