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

The product rule states that for

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

- Identify the two functions: ( u = 9-x ) and ( v = \frac{5}{x^2} - 4 ).
- Find the derivatives of ( u ) and ( v ): ( u' = -1 ) and ( v' = -\frac{10}{x^3} ).
- Apply the product rule: ( f'(x) = u'v + uv' ).
- Substitute the derivatives and the original functions into the formula: ( f'(x) = (-1)(\frac{5}{x^2} - 4) + (9-x)\left(-\frac{10}{x^3}\right) ).
- Simplify the expression: ( f'(x) = -\frac{5}{x^2} + 4 + \frac{10}{x^3} - 9 + x ).
- Combine like terms: ( f'(x) = x - 9 -\frac{5}{x^2} + \frac{10}{x^3} + 4 ).
- Arrange the terms: ( f'(x) = x - 5/x^2 + 10/x^3 - 5 ).

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