Question

How do you find the derivative of the inverse of $f\left( x \right)=7x+6$ using the definition of the derivative of an inverse function?

Answer 1

In this question we have been given a function $f\left( x \right)$ for which we have to find the derivative of its inverse function. we will first convert the given function into its inverse form which is represented as $f'\left( x \right)$, and then find the derivative of the function $f'\left( x \right)$. . Inverse of the function is the image of the function reflected over the line $y=x$. In this question, we will change the function definition by considering $f\left( x \right)=y$ and then solving for the value of $x$, which will give us the required inverse function.

Complete step-by-step solution:
We have the given function as:
$\Rightarrow f\left( x \right)=7x+6$
In this question, we will consider $f\left( x \right)=y$ and change the function definition. On substituting it in the equation, we get:
$\Rightarrow y=7x+6$
On transferring the term $7x$ from the right-hand side to the left-hand side, we get:
$\Rightarrow y-7x=6$
On transferring the term $y$ from the left-hand side to the right-hand side, we get:
$\Rightarrow -7x=-y+6$
On transferring $-7$ from left-hand side to the right-hand side, we get:
$\Rightarrow x=\dfrac{-y+6}{-7}$
On taking the negative sign common in the numerator, we get:
$\Rightarrow x=\dfrac{-\left( y-6 \right)}{-7}$
On cancelling the negative sign, we get:
$\Rightarrow x=\dfrac{y-6}{7}$
On substituting $y=x$ to get the expression in terms of $x$, we get:
$\Rightarrow f\left( x \right)=\dfrac{x-6}{7}$
Now we have to find the derivative of the given function therefore, we can write it as:
$\Rightarrow \dfrac{d}{dx}f'\left( x \right)=\dfrac{d}{dx}\dfrac{x-6}{7}$
On taking the constant $\dfrac{1}{7}$ out of the derivative, we get:
$\Rightarrow \dfrac{1}{7}\dfrac{d}{dx}\left( x-6 \right)$
Now on splitting the derivative, we get:
$\Rightarrow \dfrac{1}{7}\left( \dfrac{d}{dx}\left( x \right)-\dfrac{d}{dx}\left( 6 \right) \right)$
Now we know that $\dfrac{dx}{dx}=1$ and $\dfrac{d}{dx}k=0$ therefore, on substituting, we get:
$\Rightarrow \dfrac{1}{7}\left( 1-0 \right)$
On simplifying, we get:
$\Rightarrow \dfrac{1}{7}$, which is the required solution therefore, $\dfrac{d}{dx}f'\left( x \right)=\dfrac{1}{7}$.

Note: Inverse function is a function which reverses the value of the function. It is also called the anti-function. The basic steps to solve the inverse of a function should be remembered which is that every instance of $x$ should be replaced by $y$ and every instance of $y$ should be replaced with $x$, then solve for $y$ to get the inverse function.