How do you use the formula (ln f(x))'= f'(x) / f(x) to show that lnx and ln(2x) have the same derivative?
We are aware of:
Thus:
Thus:
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To show that (\ln(x)) and (\ln(2x)) have the same derivative using the formula ((\ln f(x))' = \frac{f'(x)}{f(x)}), we'll differentiate both functions separately.
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For (\ln(x)):
- Let (f(x) = x).
- Then, (f'(x) = 1).
- Using the formula, we get: ((\ln x)' = \frac{1}{x}).
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For (\ln(2x)):
- Let (f(x) = 2x).
- Then, (f'(x) = 2).
- Using the formula, we get: ((\ln(2x))' = \frac{2}{2x} = \frac{1}{x}).
Both derivatives are (\frac{1}{x}), showing that (\ln(x)) and (\ln(2x)) indeed have the same derivative.
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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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