Question

Find the condition for ‘ $f$ ’ if $f(x) = \dfrac{1}{{x + 1}} - \log \left( {1 + x} \right)$ where, $x > 0$ which resembles the certain condition.
(a) None of these
(b) Cannot be determined
(c) An increasing function
(d) A decreasing function

Answer 1

Hint : The given problem revolves around the concepts of derivative equations and its respective conditions such as $x > 0$ . So, first of all we will derive the given function or functions with respect to ‘ $x$ ’. Then, by using certain laws of derivatives such as for multiplication, dividation, etc. and the rules like $\dfrac{d}{{dx}}\left( {\dfrac{1}{x}} \right) = - \dfrac{1}{{{x^2}}}$ , $\dfrac{d}{{dx}}\left( {\log x} \right) = \dfrac{1}{x}$ , etc. the desired solution can be obtained.

Complete step-by-step answer :
Since, we have given the function that
$f(x) = \dfrac{1}{{x + 1}} - \log \left( {1 + x} \right)$
Where the conditions $x > 0$ exists!
As a result, derivating the given function with respect to ‘ $x$ ’ , we get
$f'(x) = - \dfrac{1}{{{{\left( {x + 1} \right)}^2}}} \times \left( 1 \right) - \dfrac{1}{{1 + x}}\left( 1 \right)$
Where as, $\dfrac{d}{{dx}}\left( {\dfrac{1}{x}} \right) = - \dfrac{1}{{{x^2}}}$ and $\dfrac{d}{{dx}}\left( {\log x} \right) = \dfrac{1}{x}$
Since, applying the certain rules of derivatives, the equation is obtained
As a result, simplifying it further, we get
$f'(x) = - \dfrac{1}{{{{\left( {x + 1} \right)}^2}}} - \dfrac{1}{{1 + x}}$
Taking ‘ $\dfrac{1}{{1 + x}}$ ’ common form the above equation, we get
$f'(x) = - \dfrac{1}{{1 + x}}\left( {\dfrac{1}{{1 + x}} + 1} \right)$
Since, from the above solution it seems that
The function earns the negative value at given condition that is $x > 0$ respectively,
And, vice-versa i.e. positive at $x < 0$ particularly
$\Rightarrow \therefore$ The option (d) is absolutely correct!
So, the correct answer is “Option d”.

Note : One must be able to derive the given equation with the respective variable being asked to solve or to find the condition (like here) using different formulae that are rules of derivatives. Also, remember the laws of derivatives mandatorily, so as to be sure of our final answer. High level of thinking is preferable in this case for the ease of the problem/s.