If #f(x)=  x^2 + x # and #g(x) = sqrtx + x #, 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:
 Identify the outer function ( f(x) ) and the inner function ( g(x) ).
 Differentiate the outer function with respect to the inner function ( g(x) ).
 Differentiate the inner function ( g(x) ) with respect to ( x ).
 Multiply the results from steps 2 and 3 to get the final derivative.
Here's the breakdown:

Outer function: ( f(x) = x^2 + x ) Inner function: ( g(x) = \sqrt{x} + x )

Differentiate ( f(x) ) with respect to ( g(x) ): ( f'(g) = 2g + 1 )

Differentiate ( g(x) ) with respect to ( x ): ( g'(x) = \frac{1}{2\sqrt{x}} + 1 )

Multiply ( f'(g) ) and ( g'(x) ): ( f'(g) \times g'(x) = (2g + 1) \times (\frac{1}{2\sqrt{x}} + 1) )

Substitute ( g(x) = \sqrt{x} + x ) into the expression obtained in step 4 to get the final derivative.
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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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