# How do you differentiate #f(x)=1/x+5sinx#?

Using the linearity of differentiation we have:

The first term can be differentiated using the power rule:

while the second term is the known derivative of a trigonometric function:

And in conclusion:

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To differentiate ( f(x) = \frac{1}{x} + 5 \sin(x) ), apply the rules of differentiation. The derivative of ( \frac{1}{x} ) is ( -\frac{1}{x^2} ), and the derivative of ( 5 \sin(x) ) is ( 5 \cos(x) ). So, the derivative of ( f(x) ) is ( f'(x) = -\frac{1}{x^2} + 5 \cos(x) ).

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To differentiate ( f(x) = \frac{1}{x} + 5 \sin(x) ), you can use the sum rule and the derivative of each term individually.

[ f'(x) = \frac{d}{dx} \left( \frac{1}{x} \right) + \frac{d}{dx} \left( 5 \sin(x) \right) ]

[ f'(x) = -\frac{1}{x^2} + 5 \cos(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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