Question

How do you find the derivative of $ \ln (\dfrac{x}{3}) $ ?

Answer 1

Hint: Differentiation, also called derivative, is a process of finding out the rate of change of a quantity with respect to some other quantity for a very short period of time. On finding out the derivative of a function, when we put the value of the unknown quantity, we get its instantaneous rate of change at that particular value. In this question, we have to find the derivative of $ \ln (\dfrac{x}{3}) $ , so we use the knowledge of differentiation of some basic functions to find out the correct answer.

Complete step-by-step answer:
We know that $ \dfrac{{d\ln x}}{{dx}} = \dfrac{1}{x} $ , and we have to find the derivative of $ \ln (\dfrac{x}{3}) $ , so –
 $
  \dfrac{{d\ln (\dfrac{x}{3})}}{{dx}} = \dfrac{1}{{\dfrac{x}{3}}}\dfrac{{d(\dfrac{x}{3})}}{{dx}}
  = \dfrac{3}{x} \times \dfrac{1}{3} \\
   \Rightarrow \dfrac{{d\ln (\dfrac{x}{3})}}{{dx}} = \dfrac{1}{x} \;
  $
Hence, the derivative of $ \ln (\dfrac{x}{3}) $ is $ \dfrac{1}{x} $ .
So, the correct answer is “ $ \dfrac{1}{x} $ ”.

Note: In daily life, we observe the phenomenon at a big scale so we consider the change over specific/long duration of time, but behind those big works, there are small scale details that one can’t ignore so it becomes important to find the change within a very short duration of time that is the instantaneous change. Differentiation is used for finding the instantaneous change, differentiation of a quantity y with respect to a quantity x is given as $ \dfrac{{dy}}{{dx}} $ where dy represents a very small change in y and dx represents a very small change in x, For example – the instantaneous speed of a body is given as $ \dfrac{{dx}}{{dt}} $ . The derivatives of different functions are different, so we have to memorize the derivatives of some basic functions to solve questions like this one.