If #f(x)= sin3x # and #g(x) = 2x^2 3x #, 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:
 Find ( f'(g(x)) ), the derivative of ( f(x) ) with respect to ( g(x) ).
 Find ( g'(x) ), the derivative of ( g(x) ) with respect to ( x ).
 Multiply ( f'(g(x)) ) by ( g'(x) ) to get the final derivative.
Here's how to apply this to the given functions:

( f(x) = \sin(3x) ) ( f'(x) = 3\cos(3x) ) (derivative of (\sin(ax)) is (a\cos(ax)))

( g(x) = 2x^2  3x ) ( g'(x) = 4x  3 ) (derivative of (2x^2) is (4x), derivative of (3x) is (3))

Now, substitute ( g(x) ) into ( f(x) ) to get ( f(g(x)) ): ( f(g(x)) = \sin(3(2x^2  3x)) = \sin(6x^2  9x) )

Apply the chain rule: ( f'(g(x)) = 3\cos(6x^2  9x) )

Multiply by ( g'(x) ): ( f'(g(x)) \cdot g'(x) = (3\cos(6x^2  9x)) \cdot (4x  3) )
So, the derivative of ( f(g(x)) ) with respect to ( x ) using the chain rule is ( (3\cos(6x^2  9x)) \cdot (4x  3) ).
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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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