How do you differentiate #f(x)=1/x+x# using the sum rule?

Answer 1

#f'(x) = -1/x^2 + 1#

The Sum Rule simply states that you take the derivative of each term and add them together.

#1/x# can be re-written as #x^-1#. This makes it clear that you want to use the Power Rule with this one. So, using the Power Rule, you bring down the #-1# from the exponent, and the exponent decreases to #-2#. #-x^-2# is written as #-1/x^2#. So, the derivative of the first term is #-1/x^2#.
The second term is easy - you should know that the derivative of x is 1. If you don't, you can apply the Power Rule again, and receive an answer of #x^0#, which is #1#.
So, when you use the Sum Rule, you add these derivatives together. The result is: #f'(x) = -1/x^2 + 1#. I hope this helped.
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Answer 2

To differentiate the function ( f(x) = \frac{1}{x} + x ) using the sum rule, you differentiate each term separately and then sum the results.

[ f(x) = \frac{1}{x} + x ] [ f'(x) = \frac{d}{dx}\left(\frac{1}{x}\right) + \frac{d}{dx}(x) ]

Using the power rule for differentiation, (\frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2}) and (\frac{d}{dx}(x) = 1).

Thus,

[ f'(x) = -\frac{1}{x^2} + 1 ]

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Answer from HIX Tutor

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.

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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