Question

If $\left[ x \right]$denotes the greatest integer contained in $x$, then for $4 < x < 5$, $\dfrac{d}{{dx}}\left\{ {\left[ x \right]} \right\} = $
1. $\left[ {x - 4,5} \right]$
2. $\left[ x \right]$
3. $0$
4. $1$

Answer 1

Hint: In this question, we are given that $\left[ x \right]$ is a greatest integer fraction where $4 < x < 5$and we have to find the first derivative of given function. Using $\left[ {n + f} \right] = n$ formula, you’ll get the value of $\left[ x \right]$ and differentiate that with respect to $x$.

Formula Used:
$\left[ {n + f} \right] = n$, $n$ is the integer and $f$ is the fraction

Complete step by step Solution:
Given that,
Let, $\left[ x \right]$be the greatest integer contained in $x$,
Also, $4 < x < 5 - - - - - \left( 1 \right)$
As we know that, $\left[ {} \right]$ denotes the greatest integer function
Where $\left[ {n + f} \right] = n$,
Here, $n$ is the integer and $f$is the fraction. So, the greatest integer value function gives the greatest integral value less than the input.
From equation (1), $x$ should be less than $5$
It implies that, $\left[ x \right] = 4$( constant)
Now, we know that the derivative of a constant with respect to a variable is equal to zero.
So, differentiating equation (2) with respect to $x$, we get,
$\dfrac{d}{{dx}}\left\{ {\left[ x \right]} \right\} = \dfrac{d}{{dx}}\left( 4 \right) = 0$

Hence, the correct option is 3.

Additional Information: The greatest integer function is one that returns an integer that is closer to the given real number. It is also known as the step function. The greatest integer function returns the nearest integer to the given number. Also, aside from integration, differentiation is one of the two most significant notions. Differentiation is known as the technique for determining a function's derivative. Basically, differentiation is a mathematical method that determines the function’s instantaneous rate of change based on only one of its variables. The most common example: is velocity, because velocity is the rate of change of displacement (distance) with respect to time and the inverse of finding a derivative is known as anti-differentiation. If $x$ is a variable and $y$ is another variable, then $\dfrac{{dy}}{{dx}}$ is the rate of change of $x$ with respect to $y$. The general expression for a function's derivative is $f'\left( x \right) = \dfrac{{dy}}{{dx}}$, where $y = f\left( x \right)$ can be any function.

Note: The greatest integer function is one that returns an integer that is closer to the given real number. It is also known as the step function. The greatest integer function returns the nearest integer to the given number.