# What is the derivative of #f(x)=x*log_5(x)# ?

Now, differentiate, and apply the product rule:

Simplifying yields:

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To find the derivative of ( f(x) = x \cdot \log_5(x) ), you can use the product rule. The product rule states that if you have two functions, ( u(x) ) and ( v(x) ), then the derivative of their product is given by ( (u \cdot v)' = u'v + uv' ).

Let ( u(x) = x ) and ( v(x) = \log_5(x) ).

The derivative of ( u(x) = x ) is ( u'(x) = 1 ).

The derivative of ( v(x) = \log_5(x) ) can be found using the chain rule:

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

Now, applying the product rule:

[ f'(x) = u'v + uv' ] [ f'(x) = 1 \cdot \log_5(x) + x \cdot \frac{1}{x \ln(5)} ] [ f'(x) = \log_5(x) + \frac{1}{\ln(5)} ]

So, the derivative of ( f(x) = x \cdot \log_5(x) ) is ( \log_5(x) + \frac{1}{\ln(5)} ).

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