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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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.
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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