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What is the $10th$ derivative ${f^{10}}(x)$of the function $f(x) = \dfrac{1}{x}$?
A. $\dfrac{{10!}}{{{x^{10}}}}$
B. $\dfrac{{ - 10!}}{{{x^{10}}}}$
C. $\dfrac{{ - 10!}}{{{x^{11}}}}$
D. $\dfrac{{10!}}{{{x^{11}}}}$

Last updated date: 14th Jul 2024
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Hint:We are given the function, we have to find its tenth derivative, the derivative of the function which is in the fraction can be found out by first making it into the numerator from the denominator by making its power negative, then using the standard formulae for differentiation.

Complete step by step answer:
The formula of a variable with exponential power is $\dfrac{d}{{dx}}({x^n}) = n{x^{n - 1}}$.This formula will help us to find its derivative, since the tenth derivative is asked so we will first find the derivative upto the second or third time, then generalize it to get the standard formula and then put it to get the value of the tenth derivative of the given function.

The derivative for a function with an index is given by the formula,
$\dfrac{d}{{dx}}({x^n}) = n{x^{n - 1}}$
We are given the function,
$\dfrac{1}{x}$, solving it we will get,
${x^{ - 1}}$, this form can now be easily differentiated,
The differentiation of it is given by,
$ \Rightarrow \dfrac{d}{{dx}}({x^{ - 1}}) = - {x^{ - 2}}$
The second derivative will be,
$ \Rightarrow {\dfrac{d}{{dx}}^2}({x^{ - 1}}) = 2{x^{ - 2}}$

The third derivative will be given by,
$ \Rightarrow {\dfrac{d}{{dx}}^3}({x^{ - 1}}) = - 6{x^{ - 3}}$,
 thus we can see the general trend that,
When $n$ is even the derivative is given by,
${\dfrac{d}{{dx}}^n}({x^{ - 1}}) = {( - 1)^n}n!{x^{ - n}}$
This means that the odd term will be negative and the even term will be positive,
Thus according to this trend we can say the tenth derivative of the given function will be,
$ \Rightarrow {\dfrac{d}{{dx}}^{10}}({x^{ - 1}}) = {( - 1)^{10}}10!{x^{ - 10}}$
Solving which we get,
$ \therefore {\dfrac{d}{{dx}}^{10}}({x^{ - 1}}) = 10!\dfrac{1}{{{x^{10}}}}$

Hence, the correct answer is option A.

Note:The symbol $!$ here is the factorial sign which means we have to find the factorial of the number, this is why we have written the general formula in terms of $n!$, it is also necessary to remember that the factorial of $0$ is $1$ not $0$, also the term ${( - 1)^n}$ helps us to get positive values at even intervals and negative values at odd intervals.