Question

What is the derivative of $6\ln \left( x \right)$?

Answer 1

To obtain the answer of this type of question use differentiation of logarithm function formula. Firstly check whether there is any need to simplify the function more or is it in its general form. Then use the formula $\dfrac{d\left( \log x \right)}{dx}=\dfrac{1}{x}\times \dfrac{d\left( x \right)}{dx}$ to find the derivative of the function. Finally simplify it to get the desired answer.

Complete step-by-step answer:
To find the derivative of $6\ln \left( x \right)$ with respect to x we will be using the differentiation formula of logarithm.
So as we know that differentiation of logarithm function is done as follows:
$\dfrac{d\left( \log x \right)}{dx}=\dfrac{1}{x}\times \dfrac{d\left( x \right)}{dx}$………$\left( 1 \right)$
To find derivative of $6\ln \left( x \right)$ we will use formula (1) as follows:
$\begin{aligned} & \Rightarrow \dfrac{d\left( 6\ln \left( x \right) \right)}{dx}=6\dfrac{d\left( \ln \left( x \right) \right)}{dx} \\\ & \Rightarrow \dfrac{d\left( 6\ln \left( x \right) \right)}{dx}=6\times \dfrac{1}{x} \\\ & \therefore \dfrac{d\left( 6\ln \left( x \right) \right)}{dx}=\dfrac{6}{x} \\\ \end{aligned}$
So the derivative is $\dfrac{6}{x}$
Hence derivative of $6\ln \left( x \right)$ is $\dfrac{6}{x}$

Note: The process to find the derivative of a function is known as differentiation. It is used to find the instantaneous rate of change in a function which depends on one of its variables. The logarithm function is an inverse function of the exponential function. In general it is defined as $y={{\log }_{b}}x$ where $b$ is the base and it is read as “log base $b$ of $x$”. If the base is not given we consider it as $e$ which is known as natural logarithm. The differentiation of logarithm function is different from the differentiation of exponential function as exponential function when differentiated gives the same function with its power multiplied to it. In logarithm function differentiation the term inside the bracket is taken in the denominator multiplied by the differentiation of that term. In some cases logarithm identities are also used to simplify the calculation.