Question

How do you find the derivatives of $\ln \left( {\dfrac{{10}}{x}} \right)$ ?

Answer 1

In this question, we have to differentiate $\ln \left( {\dfrac{{10}}{x}} \right)$. Here $\ln$ represents the natural logarithm to the base $e$. We first simplify the given term by using properties of logarithm. The given term is of the form $\log \left( {\dfrac{a}{b}} \right) = \log a - \log b$. After that we differentiate each term to obtain the required result.

Complete step by step answer:
Given the function, $\ln \left( {\dfrac{{10}}{x}} \right)$ …… (1)
We are asked to find the derivative of the function given in the equation (1).
The given function is a logarithmic function. The logarithmic function is represented as ${\log _b}a$, where b is called the base and a is a number.
Here we have given the natural logarithmic function where it’s base is $e$ and it is represented as $\ln$.
Now we make use of some basic properties of logarithmic function to evaluate it.
Note that the given function is of the form of $\ln \left( {\dfrac{a}{b}} \right)$.
We have the property related to it and it is given as $\ln \left( {\dfrac{a}{b}} \right) = \ln a - \ln b$.
Note that here $a = 10$ and $b = x$.
Now applying the property, we have,
$\ln \left( {\dfrac{{10}}{x}} \right) = \ln 10 - \ln x$ …… (2)
Now we differentiate the above function to obtain the required derivative.
Differentiating equation (2), we get,
$\dfrac{d}{{dx}}\ln \left( {\dfrac{{10}}{x}} \right) = \dfrac{d}{{dx}}\left( {\ln 10 - \ln x} \right)$
Now differentiating each term in the L.H.S. we get,
$\Rightarrow \dfrac{d}{{dx}}\ln \left( {\dfrac{{10}}{x}} \right) = \dfrac{d}{{dx}}(\ln 10) - \dfrac{d}{{dx}}(\ln x)$
We know that the derivative of a constant term is equal to zero.
Since $\ln 10$ is a constant, we get,
$\dfrac{d}{{dx}}(\ln 10) = 0$
Hence we get,
$\Rightarrow \dfrac{d}{{dx}}\ln \left( {\dfrac{{10}}{x}} \right) = 0 - \dfrac{d}{{dx}}(\ln x)$
$\Rightarrow \dfrac{d}{{dx}}\ln \left( {\dfrac{{10}}{x}} \right) = - \dfrac{d}{{dx}}(\ln x)$
We know that $\dfrac{d}{{dx}}\ln x = \dfrac{1}{x}$
Substituting this value we get,
$\Rightarrow \dfrac{d}{{dx}}\ln \left( {\dfrac{{10}}{x}} \right) = - \dfrac{1}{x}$

Hence, the derivative of the function $\ln \left( {\dfrac{{10}}{x}} \right)$ is $- \dfrac{1}{x}$.

Note: If the question has the word log or $\ln$, it represents the given function as logarithmic function. Note that we have two types of logarithmic function.
One is a common logarithmic function which is represented as a log and its base is 10.
The other one is a natural logarithmic function represented as $\ln$ and it’s base is $e$.
We must know the basic properties of logarithmic functions and note that these properties hold for both log and $\ln$functions.
Remember the differentiation of logarithmic function, $\dfrac{d}{{dx}}\log x = \dfrac{1}{x}$.
Some properties of logarithmic functions are given below.
(1) $\ln (x \cdot y) = \ln x + \ln y$
(2) $\ln \left( {\dfrac{x}{y}} \right) = \ln x - \ln y$
(3) $\ln {x^n} = n\ln x$
(4) $\ln 1 = 0$
(5) ${\log _e}e = 1$