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

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Answer 1 · 2014-08-06T14:51:19

When you're differentiating an exponential with a base other than #e#, use the change-of-base rule to convert it to natural logarithms:

#f(x) = x * lnx/ln5#

Now, differentiate, and apply the product rule:

#d/dxf(x) = d/dx[x] * lnx/ln5 + x * d/dx[lnx/ln5]#

We know that the derivative of #ln x# is #1/x#. If we treat #1/ln5# as a constant, then we can reduce the above equation to:

#d/dxf(x) = lnx/ln5 + x/(xln5)#

Simplifying yields:

#d/dxf(x) = (lnx+1)/ln5#