Question

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

Answer 1

Hint: In order to find the first derivative of the above expression with respect to x Use the reciprocal rule of derivation i.e. $ {\left[ {\dfrac{1}{{u(x)}}} \right] ^\prime } = \dfrac{{u'(x)}}{{u{{(x)}^2}}} $ ,considering $ u\left( x \right) = 1 - x $ 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 $
 $ \dfrac{d}{{dx}}\left[ 1 \right] = 0 $

Complete step-by-step answer:
 Given a function $ \dfrac{1}{{1 - x}} $ let it be $ f(x) $
 $ f(x) = \dfrac{1}{{1 - x}} $
We have to find the first derivative of the above equation
 \[
  \dfrac{d}{{dx}}\left[ {f(x)} \right] = f'(x) \\
  f'(x) = \dfrac{d}{{dx}}\left[ {\dfrac{1}{{1 - x}}} \right] \;
 \]
Differentiation is linear So we can differentiate summands easily and pull out the constant factors
  \[f'(x) = \dfrac{d}{{dx}}\left[ {\dfrac{1}{{1 - x}}} \right] \]
Let’s assume $ u\left( x \right) = 1 - x $ and applying the reciprocal rule $ {\left[ {\dfrac{1}{{u(x)}}} \right] ^\prime } = \dfrac{{u'(x)}}{{u{{(x)}^2}}} $
 \[ = - \dfrac{{\dfrac{d}{{dx}}\left[ 1 \right] - \dfrac{d}{{dx}}\left[ x \right] }}{{{{(1 - x)}^2}}}\]
The derivative of differentiation variable is 1
And derivative of the constant is 0
 $
   = - \dfrac{{0 - 1}}{{{{(1 - x)}^2}}} \\
   = \dfrac{1}{{{{(1 - x)}^2}}} \;
  $
Therefore, the derivative of $ \dfrac{1}{{1 - x}} $ is equal to $ \dfrac{1}{{{{(1 - x)}^2}}} $ or $ {(1 - x)^{ - 2}} $ .
So, the correct answer is “ $ \dfrac{1}{{{{(1 - x)}^2}}} $ or $ {(1 - x)^{ - 2}} $ ”.

Note: In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus.