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.