# How do you differentiate #y= log _5 x#?

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To differentiate ( y = \log_5 x ), you can use the logarithmic differentiation rule.

First, express ( y ) in terms of ( \ln ) (natural logarithm) to make the differentiation easier:

[ y = \frac{\ln x}{\ln 5} ]

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

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

Apply the quotient rule:

[ \frac{dy}{dx} = \frac{\frac{d}{dx}(\ln x) \cdot \ln 5 - \ln x \cdot \frac{d}{dx}(\ln 5)}{(\ln 5)^2} ]

Now, find the derivatives:

[ \frac{dy}{dx} = \frac{\frac{1}{x} \cdot \ln 5 - 0}{(\ln 5)^2} ]

[ \frac{dy}{dx} = \frac{\ln 5}{x \cdot (\ln 5)^2} ]

Thus, the derivative of ( y = \log_5 x ) is:

[ \frac{dy}{dx} = \frac{\ln 5}{x \cdot (\ln 5)^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.

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