# How do you differentiate #g(z)=(z^2+1)/(x^3-5)# using the quotient rule?

The quotient rule can be stated as;

Now we have all of the pieces we need, we can plug them into our power rule function.

Now we can simplify our terms.

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To differentiate ( g(z) = \frac{{z^2 + 1}}{{z^3 - 5}} ) using the quotient rule:

- Identify ( f(z) = z^2 + 1 ) and ( h(z) = z^3 - 5 ).
- Apply the quotient rule:

[ g'(z) = \frac{{h(z) \cdot f'(z) - f(z) \cdot h'(z)}}{{[h(z)]^2}} ]

- Find the derivatives of ( f(z) ) and ( h(z) ):

[ f'(z) = 2z ] [ h'(z) = 3z^2 ]

- Plug the derivatives and original functions into the quotient rule formula:

[ g'(z) = \frac{{(z^3 - 5)(2z) - (z^2 + 1)(3z^2)}}{{(z^3 - 5)^2}} ]

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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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