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How do you differentiate #g(x)=(3-1/x)/(x+5)# using the quotient rule?

Answer 1

Isaiah Allen

I would change the form first.

Multiply by #x/x# to get

#g(x) = (3x-1)/(x^2+5x)#

So,

#g'(x) = ((3)(x^2+5x)-(3x-1)(2x+5))/(x^2+5x)^2#

# = (3x^2+15x-6x^2-15x+2x+5)/(x^2+5x)^2#

# = (-3x^2+2x+5)/(x^2+5x)^2#

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Answer 2

Samantha Adkison

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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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.