Question

How do you find the derivative of $ \dfrac{1}{{2x}} $ ?

Answer 1

Hint: Use the reciprocal rule of derivation i.e. $ {\left[ {\dfrac{1}{{u(x)}}} \right]^\prime } = \dfrac{{u'(x)}}{{u{{(x)}^2}}} $ to solve the above problem .
Formula:
 $ {\left[ {\dfrac{1}{{u(x)}}} \right]^\prime } = \dfrac{{u'(x)}}{{u{{(x)}^2}}} $
 $ \dfrac{d}{{dx}}\left[ x \right] = 1 $

Complete step-by-step answer:
 Given a function $ \dfrac{1}{{2x}} $ let it be $ f(x) $
 $ f(x) = \dfrac{1}{{2x}} $
We have to find the first derivative of the above equation
 $
  \dfrac{d}{{dx}}\left[ {f(x)} \right] = f'(x) \\
  f'(x) = \dfrac{d}{{dx}}\left[ {\dfrac{1}{{2x}}} \right] \\
  $
Differentiation is linear So we can differentiate summands easily and pull out the constant factors
  $ f'(x) = \dfrac{1}{2}\dfrac{d}{{dx}}\left[ {\dfrac{1}{x}} \right] $
Now applying the reciprocal rule $ {\left[ {\dfrac{1}{{u(x)}}} \right]^\prime } = \dfrac{{u'(x)}}{{u{{(x)}^2}}} $
 $ = - \dfrac{{\dfrac{{\dfrac{d}{{dx}}\left[ x \right]}}{{{x^2}}}}}{2} $
The derivative of differentiation variable is 1
 $ = - \dfrac{1}{{2{x^2}}} $
So, the correct answer is “- $ \dfrac{1}{{2{x^2}}} $ ”.

Note: 1. Calculus consists of two important concepts one is differentiation and other is integration.
2.What is Differentiation?
It is a method by which we can find the derivative of the function .It is a process through which we can find the instantaneous rate of change in a function based on one of its variables.
Let y = f(x) be a function of x. So the rate of change of $ y $ per unit change in $ x $ is given by:
 $ \dfrac{{dy}}{{dx}} $ .