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

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Answer 1 · 2016-01-09T15:58:21

$1/x$

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

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