How do you use the formula (ln f(x))'= f'(x) / f(x) to show that lnx and ln(2x) have the same derivative?

Answer 1

We are aware of:

#(lnf(x))'=(f'(x))/f(x)#
So, for #lnx#, we see that #f(x)=x#, which implies that #f'(x)=1#.

Thus:

#(lnx)'=1/x#
For #ln(2x)#, we see that #f(x)=2x#, so #f'(x)=2#.

Thus:

#(ln(2x))'=2/(2x)=1/x#
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Answer 2

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.

  1. For (\ln(x)):

    • Let (f(x) = x).
    • Then, (f'(x) = 1).
    • Using the formula, we get: ((\ln x)' = \frac{1}{x}).
  2. 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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Answer from HIX Tutor

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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