Question

What is the derivative of \[2\ln (x)\] \[?\]

Answer 1

Hint: Note that the derivative of the given function \[2\ln (x)\] can be obtained by another method using the property ‘the derivative of the constant times the function is equal to constant times the derivative of the given function’.

Complete step by step solution:
Since we know that the derivative of the constant function is equal to zero. Also \[\dfrac{{d\ln (x)}}{{dx}} = \dfrac{1}{x}\] .
Suppose \[u\] and \[v\] are two functions. Then differentiation has the following rules.
1. \[\dfrac{d}{{dx}}\left( {u \times v} \right) = v \times \dfrac{d}{{dx}}\left( u \right) + u \times \dfrac{d}{{dx}}\left( v \right)\] .
Differentiating with respect to \[x\] both sides of the equation (1), we get
 \[\dfrac{d}{{dx}}\left( {2\ln (x)} \right) = \ln (x) \times \dfrac{d}{{dx}}\left( 2 \right) + 2 \times \dfrac{d}{{dx}}\left( {\ln (x)} \right)\] \[ = 0 + \dfrac{2}{x} = \dfrac{2}{x}\] .
Hence, the derivative of the given function \[2\ln (x)\] is \[\dfrac{2}{x}\].

Note: A differential equation is the equation which contains dependent variables, independent variables and derivatives of the dependent variables with respect to the independent variables. Since differential equations are classified into two types, Ordinary differential equations where dependent variables depend on only one independent variable and Partial differential equations where dependent variables depend on two or more independent variables.
The order of the given differential equation is the order of the highest derivative involved in the given differential equation.