If #f(x)= 2 x^2 + x # and #g(x) = 2e^x + 1 #, how do you differentiate #f(g(x)) # using the chain rule?
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See below.
Using the chain rule:
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To differentiate ( f(g(x)) ) using the chain rule, first find ( f'(x) ) and ( g'(x) ), then substitute ( g(x) ) into ( f'(x) ) and multiply by ( g'(x) ).
Given ( f(x) = 2x^2 + x ) and ( g(x) = 2e^x + 1 ):

Find ( f'(x) ): ( f'(x) = 4x + 1 )

Find ( g'(x) ): ( g'(x) = 2e^x )

Substitute ( g(x) ) into ( f'(x) ): ( f'(g(x)) = 4g(x) + 1 )

Multiply by ( g'(x) ): ( f'(g(x)) \cdot g'(x) = (4g(x) + 1) \cdot 2e^x )
Therefore, the derivative of ( f(g(x)) ) using the chain rule is ( (4g(x) + 1) \cdot 2e^x ).
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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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