Question

How do you find the derivative of $y = \ln \left( {5x} \right)$?

Answer 1

In the given question, we have been given a logarithmic function. It has a base of Euler’s number, hence is the natural log. It has a variable multiplied with a constant as its argument. We have to find the derivative of the given function. To do that, we need to know the function proper derivative of the given function.

Formula Used:
We are going to use a basic property of log:
$\log \left( {a \times b} \right) = \log a + \log b$

Complete step-by-step answer:
The given function is $y = \ln \left( {5x} \right)$.
First, we are going to use a basic property of log:
$\log \left( {a \times b} \right) = \log a + \log b$
Hence, $y = \ln \left( 5 \right) + \ln \left( x \right)$
Now, $\dfrac{{dy}}{{dx}} = \dfrac{{d\left( {\ln 5} \right)}}{{dx}} + \dfrac{{d\left( {\ln x} \right)}}{{dx}}$
But $\ln \left( 5 \right)$ is a constant, hence its derivative is zero.
Thus, $\dfrac{{dy}}{{dx}} = \dfrac{{d\left( {\ln 5} \right)}}{{dx}} + \dfrac{{d\left( {\ln x} \right)}}{{dx}} = 0 + \dfrac{1}{x} = \dfrac{1}{x}$
Hence, the derivative of $\ln \left( {5x} \right)$ is $\dfrac{1}{x}$.

Additional Information:
The $\log$ function has other basic properties too:
${\log _b}a = n \Rightarrow {b^n} = a$
${\log _a}b = \dfrac{1}{{{{\log }_b}a}}$
${\log _x}{x^n} = n$

Note: In the given question, we had to find the derivative of a natural log function. For finding that, we only need to know the result we get after differentiating the natural log function, which is a standard result. So, it is really important that we know the formulae and where, when and how to use them so that we can get the correct result.