How do you find the derivative of #f(x)=Log_5(x)#?

Answer 1

#1/(xln(5))#

We want to get this into natural logs as we know how to differentiate them.

#y = log_5x#
#5^y = x#
#ln(x) = ln(5^y)#
#ln(x) = yln(5) implies y = ln(x)/ln(5)#

So now we have to find

#d/(dx) ((ln(x))/(ln(5)))#
#= 1/(ln(5)) d/(dx)(ln(x))#
#= 1/(ln(5)) * 1/x = 1/(xln(5))#
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Answer 2

To find the derivative of ( f(x) = \log_5(x) ), you can use the logarithmic differentiation technique. The derivative is ( f'(x) = \frac{1}{x \ln(5)} ).

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Answer 3

To find the derivative of ( f(x) = \log_5(x) ), we can use the logarithmic differentiation technique. First, we express the function in terms of natural logarithms using the change of base formula:

[ \log_5(x) = \frac{\ln(x)}{\ln(5)} ]

Now, we can differentiate both sides of the equation with respect to ( x ):

[ \frac{d}{dx}(\log_5(x)) = \frac{d}{dx}\left(\frac{\ln(x)}{\ln(5)}\right) ]

Using the quotient rule for differentiation, we get:

[ \frac{d}{dx}(\log_5(x)) = \frac{1}{\ln(5)} \cdot \frac{d}{dx}(\ln(x)) ]

The derivative of ( \ln(x) ) with respect to ( x ) is ( \frac{1}{x} ), so substituting that in:

[ \frac{d}{dx}(\log_5(x)) = \frac{1}{\ln(5)} \cdot \frac{1}{x} ]

Therefore, the derivative of ( f(x) = \log_5(x) ) is:

[ f'(x) = \frac{1}{x \cdot \ln(5)} ]

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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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