# If #f(x)= 2 x^2 + x # and #g(x) = sqrtx + 1 #, how do you differentiate #f(g(x)) # using the chain rule?

now square out the brackets and collect like terms.

differentiating to obtain.

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To differentiate ( f(g(x)) ) using the chain rule, follow these steps:

- Substitute ( g(x) ) into ( f(x) ) to get ( f(g(x)) ).
- Differentiate ( f(g(x)) ) with respect to ( x ) using the chain rule.
- Replace ( g(x) ) with ( \sqrt{x} + 1 ).
- Differentiate ( f(x) ) with respect to ( x ).
- Multiply the result of step 4 by the derivative of ( g(x) ) with respect to ( x ).
- Combine the results to get the final derivative.

Let's go through each step:

- ( f(g(x)) = 2(\sqrt{x} + 1)^2 + (\sqrt{x} + 1) )
- ( f'(g(x)) = 4(\sqrt{x} + 1) + 1 )
- ( g'(x) = \frac{1}{2\sqrt{x}} )
- ( f'(x) = 4x + 1 )
- ( f'(g(x)) \cdot g'(x) = (4(\sqrt{x} + 1) + 1) \cdot \frac{1}{2\sqrt{x}} )
- Combine the results: ( \frac{4(\sqrt{x} + 1) + 1}{2\sqrt{x}} )

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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.

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