Question

How can I find the derivative of the inverse of \[f(x) = {x^3} + x + 1\] at \[x = 11\] .

Answer 1

Hint: We will do the derivation of the given term and then we will put this value into the inverse of the function. On doing some simplification we get the required answer.

Formula Used:
If a function is represented as \[f\] then its inverse function will be written as \[{f^{ - 1}}\] .
If a variable \[x\] is independent in the function then it will become a dependent variable in its inverse function.
Let's say, \[y = f(x)\] .
Then we can say that the inverse of this function will become:
\[x = {f^{ - 1}}(y)\] .
Now, if we take the derivation of \[x\] with respect to \[y\] , \[\dfrac{{dx}}{{dy}}\] , then its inverse will become as following:
 \[\dfrac{1}{{\dfrac{{dy}}{{dx}}}}\] .
So, we can put this into Leibnitz inverse function in following manner:
\[({f^{ - 1}})'(y) = \dfrac{1}{{f'({f^{ - 1}}(y))}}\] , as the above iteration tells us that the inverse of a function is the reciprocal of that function.
General formula of the integration of \[{x^n}\] with respect to \[x\] is:
\[\dfrac{d}{{dx}}({x^n}) = n.{x^{n - 1}}.\]

Complete step by step answer:
Let say,
\[y = f(x) = {x^3} + x + 1\] .
So, we can say that:
\[x = {f^{ - 1}}(y)\] .
So, the derivation of the above function is:
\[\dfrac{{dy}}{{dx}} = f'(x) = 3{x^2} + 1.\]
So, using the trial and error, we can show that the value of \[f(2) = {2^3} + 2 + 1 = 11\] so that the inverse of it will be \[{f^{ - 1}}(11) = 2\] .
So, we can find the value of \[{f^{ - 1}}\] at the point \[2\] .
So, we can derive the following equation for the inverse function:
\[{\left( {({f^{ - 1}})'(11)} \right)_{x = 2}} = \dfrac{1}{{f'({x^3} + x + 1)}}\]
By doing the further simplification, we get:
\[{\left( {({f^{ - 1}})'(11)} \right)_{x = 2}} = \dfrac{1}{{3{x^2} + 1}}\]
On putting the values and we get,
\[ \Rightarrow {\left( {({f^{ - 1}})'(11)} \right)_{x = 2}} = \dfrac{1}{{3 \times {2^2} + 1}}\]
On simplify the term and we get
\[ \Rightarrow {\left( {({f^{ - 1}})'(11)} \right)_{x = 2}} = \dfrac{1}{{13}}\] .

So, the value of the derivative of the above question will be \[\dfrac{1}{{13}}\] .

Note: Points to remember: If we inverse any function then the value of the independent variable becomes the dependent variable in the inverse function.