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:

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

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