# How do you find the derivative of #(5-x)/x# using limits?

The definition of derivative is

Manipulate the numerator to get

so,

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To find the derivative of (\frac{5-x}{x}) using limits, follow these steps:

- Start with the given function: (f(x) = \frac{5-x}{x}).
- Use the quotient rule: (f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}), where (u(x) = 5 - x) and (v(x) = x).
- Find the derivatives of (u(x)) and (v(x)): (u'(x) = -1) and (v'(x) = 1).
- Plug the derivatives and original functions into the quotient rule formula.
- Simplify the expression.

Applying the quotient rule:

[f'(x) = \frac{(-1)(x) - (5-x)(1)}{x^2}]

[= \frac{-x - (5-x)}{x^2}]

[= \frac{-x - 5 + x}{x^2}]

[= \frac{-5}{x^2}]

So, the derivative of (\frac{5-x}{x}) with respect to (x) using limits is (-\frac{5}{x^2}).

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