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

Aug 6, 2014

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

$f \left(x\right) = x \cdot \ln \frac{x}{\ln} 5$

Now, differentiate, and apply the product rule:

$\frac{d}{\mathrm{dx}} f \left(x\right) = \frac{d}{\mathrm{dx}} \left[x\right] \cdot \ln \frac{x}{\ln} 5 + x \cdot \frac{d}{\mathrm{dx}} \left[\ln \frac{x}{\ln} 5\right]$

We know that the derivative of $\ln x$ is $\frac{1}{x}$. If we treat $\frac{1}{\ln} 5$ as a constant, then we can reduce the above equation to:

$\frac{d}{\mathrm{dx}} f \left(x\right) = \ln \frac{x}{\ln} 5 + \frac{x}{x \ln 5}$

Simplifying yields:

$\frac{d}{\mathrm{dx}} f \left(x\right) = \frac{\ln x + 1}{\ln} 5$