Question

Find the derivative of \[{\log _{10}}x\] with respect to \[x\].

Answer 1

Hint: We need to find the derivative of \[{\log _{10}}x\] with respect to \[x\]. First of all, we will simplify the given term using properties of logarithm. Using the property \[{\log _a}b = \dfrac{{\log b}}{{\log a}}\], we will first simplify the given expression i.e. \[{\log _{10}}x\]. After that, we will differentiate it with respect to \[x\] using the derivative formulas. We know, \[\dfrac{d}{{dx}}\left( {\log x} \right) = \dfrac{1}{x}\] and \[\dfrac{d}{{dx}}\left( {c \times f(x)} \right) = c\left( {\dfrac{d}{{dx}}\left( {f(x)} \right)} \right)\] and so using these formulas, we will solve our problem.

Complete answer: We need to differentiate \[{\log _{10}}x\] with respect to \[x\] i.e. we need to find \[\dfrac{d}{{dx}}\left( {{{\log }_{10}}x} \right) - - - - - (1)\].
For that, we will first simplify \[{\log _{10}}x\].
Using the property \[{\log _a}b = \dfrac{{\log b}}{{\log a}}\], we can write
\[{\log _{10}}x = \dfrac{{\log x}}{{\log 10}}\], where \[\log 10\] is a constant term.
\[{\log _{10}}x = \dfrac{1}{{\log 10}} \times \log x - - - - - (2)\]
Hence, using (1) and (2), we have
\[\dfrac{d}{{dx}}\left( {{{\log }_{10}}x} \right) = \dfrac{d}{{dx}}\left( {\dfrac{1}{{\log 10}} \times \log x} \right)\]
So, \[{\log _{10}}x = \dfrac{1}{{\log 10}} \times \log x\] is of the form \[c \times f(x)\].
Using the formula \[\dfrac{d}{{dx}}\left( {c \times f(x)} \right) = c \times \left( {\dfrac{d}{{dx}}\left( {f(x)} \right)} \right)\], we have
\[\dfrac{d}{{dx}}\left( {{{\log }_{10}}x} \right) = \dfrac{d}{{dx}}\left( {\dfrac{1}{{\log 10}} \times \log x} \right)\]
\[ = \dfrac{1}{{\log 10}} \times \left( {\dfrac{d}{{dx}}\left( {\log x} \right)} \right)\]
Now, using the formula \[\dfrac{d}{{dx}}\left( {\log x} \right) = \dfrac{1}{x}\], we have
\[\dfrac{d}{{dx}}\left( {{{\log }_{10}}x} \right) = \dfrac{d}{{dx}}\left( {\dfrac{1}{{\log 10}} \times \log x} \right)\]
\[ = \dfrac{1}{{\log 10}} \times \left( {\dfrac{d}{{dx}}\left( {\log x} \right)} \right)\]
\[ = \dfrac{1}{{\log 10}} \times \dfrac{1}{x}\]
\[ = \dfrac{1}{{x\log 10}}\]
Therefore, we have \[\dfrac{d}{{dx}}\left( {{{\log }_{10}}x} \right) = \dfrac{1}{{x\log 10}}\].
Hence, the derivative of \[{\log _{10}}x\] with respect to \[x\] is \[\dfrac{1}{{x\log 10}}\].

Note:
We cannot directly just solve these types of questions. Firstly, we need to be very thorough with the logarithm properties. Also, while applying the properties, we need to make sure that we are applying the right property according to the given conditions. While differentiating, we should be careful as we usually get confused between integration and differentiation formulas. While we use the property \[{\log _a}b = \dfrac{{\log b}}{{\log a}}\], we consider the logarithmic base to be \[e\] if anything is not given.