Question

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

Answer 1

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`