# How do you differentiate #f(x)= (9-x)(5/(x^2 -4))# using the product rule?

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Apply product rule,

Differentiate inner terms,

Simplify,

Common denominator,

Simplify,

Combine,

Simplify,

Factorise,

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To differentiate ( f(x) = (9 - x)\left(\frac{5}{{x^2 - 4}}\right) ) using the product rule, follow these steps:

- Identify the two functions being multiplied: ( u(x) = 9 - x ) and ( v(x) = \frac{5}{{x^2 - 4}} ).
- Compute the derivatives of ( u(x) ) and ( v(x) ): ( u'(x) = -1 ) and ( v'(x) = \frac{{-10x}}{{(x^2 - 4)^2}} ).
- Apply the product rule: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
- Substitute the values into the product rule formula: ( f'(x) = (-1)\left(\frac{5}{{x^2 - 4}}\right) + (9 - x)\left(\frac{{-10x}}{{(x^2 - 4)^2}}\right) ).
- Simplify the expression to get the derivative of ( f(x) ): ( f'(x) = -\frac{5}{{x^2 - 4}} - \frac{{(9 - x)(10x)}}{{(x^2 - 4)^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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