Question

How do I find the derivative of `15/2+ln(5x)`?

Answer 1

`1/x`

Explanation:

First, notice that differentiating the `15/2` will just give `0`, so this question is identical to just finding the derivative of `ln(5x)`.

In order to differentiate functions with the natural logarithm, it's necessary to know that `d/dx(ln(x))=1/x`.

Then, through the chain rule, this can be generalized to say that `d/dx(ln(u))=1/u*u'`.

Thus,

`d/dx(ln(5x))=1/(5x)*d/dx(5x)`

Since `d/dx(5x)=5`, the derivative of the original function is

`1/(5x)*5=1/x`

Notice that this derivative is the exact same as the derivative of just `ln(x)`. This can be explained using logarithm rules:

`ln(5x)=ln(5)+ln(x)`

So, `d/dx(ln(5)+ln(x))=0+1/x=1/x`