How do you differentiate #g(x)=(31/x)/(x+5)# using the quotient rule?
I would change the form first.
So,
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To differentiate the function g(x) = (3  1/x)/(x + 5) using the quotient rule, follow these steps:

Identify the numerator and denominator functions: Numerator: 3  1/x Denominator: x + 5

Apply the quotient rule: g'(x) = (denominator * derivative of numerator  numerator * derivative of denominator) / (denominator)^2

Differentiate the numerator and denominator separately: Derivative of numerator: (1/x^2) Derivative of denominator: 1

Substitute the derivatives into the quotient rule formula: g'(x) = ((x + 5) * ((1/x^2))  (3  1/x) * 1) / (x + 5)^2

Simplify the expression: g'(x) = ((x + 5) * (1/x^2)  (3  1/x)) / (x + 5)^2

Further simplify if needed.
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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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