Answer

Verified

348k+ views

**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.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How many crores make 10 million class 7 maths CBSE

The 3 + 3 times 3 3 + 3 What is the right answer and class 8 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE