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

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

- Differentiate ( \sin(x) ) with respect to ( x ): ( \frac{d}{dx}(\sin(x)) = \cos(x) )
- Differentiate ( \frac{1}{x \cos(x)} ) with respect to ( x ) using the quotient rule: [ \frac{d}{dx}\left(\frac{1}{x \cos(x)}\right) = \frac{0 \cdot x \cos(x) - 1 \cdot (1 \cdot \cos(x) - x \cdot (-\sin(x)))}{(x \cos(x))^2} ] Simplify the expression: [ = -\frac{\cos(x) + x \sin(x)}{x^2 \cos^2(x)} ]
- Add the results from steps 1 and 2: [ f'(x) = \cos(x) - \frac{\cos(x) + x \sin(x)}{x^2 \cos^2(x)} ] Combine the terms: [ f'(x) = \frac{x^2 \cos^2(x) \cos(x) - \cos(x) - x \sin(x)}{x^2 \cos^2(x)} ] Simplify the expression if necessary.

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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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