Find the derivative of the inverse of the following functions....

Question

Find the derivative of the inverse of the following functions. Express the result with x as the independent variable.

f(x) = x^-1/3

Accepted Answer

Hello. In this video, we are going to be finding the derivative of the inverse function F X equal to X rates the power of negative. We will express the result with X as the independent variable. So, how can we find the derivative of the inverse function of the given F of X function? Well, we have an equation, an identity that can help us do this. That identity is as follows. F inverse of X is equal to 1 divided by F F inverse of X. So, just to break down what this is saying, If we want to find the derivative of the inverse function, then we are going to take one and divide this by the function F with F inverse of X plugged into it. So, there are 2 things we need to solve for in order to find the derivative of the inverse function. We need to find the derivative F of X, and we need to calculate the inverse F inverse of X. Let's go ahead and start by calculating the inverse. So, our function is F of X equal to X raises the power of -12. Now, in order to calculate the inverse, the first thing we are going to do is replace the F of X value with Y. So, we have Y is equal to X raises the power of negative/. The next thing we are going to do is we are going to swap the positions of X and Y. That is going to give us X is equal to Y raise to the power of -12. Now what we want to do is we want to go ahead and solve for why. Now, in order to solve for Y, let's go ahead and rewrite Y as a positive exponent. We can do this by placing Y under 1, that will give us X is equal to 1 divided by the square root of Y. What we will do now is we will multiply both sides of the equation by the square root of Y. That will leave us with square root of y. Multiplied by X. Equal to one. We will divide both sides of this equation by X, leaving us with square root of Y equal to 1 divided by X. And in order to eliminate the square root, we will square both sides of this equation, leaving us with Y as equal to 1 divided by X squared. This is going to be our inverse F inverse of X. Now that we have the inverse, let's go ahead and calculate the derivative. Now, we're going to calculate the derivative with respect to X. That is going to be the derivative of X rates the power of -12. That is going to give us -1 divided by 2, multiply by X raises the power of -3 divided by 2 by the power rule. Now that we have the F function, and we have the inverse function, we can use the identity to find the derivative of the inverse function. That is going to be F inverse of X equal to 1 divided. By F inverse of X plugged into F of X. That is going to be negative 1 divided by 2 multiplied by 1 divided by X2, raised to the power of -3 divided by 2. Now we just need to simplify the derivative. We can rewrite 1 divided by X2d as a negative exponent. That will give us F inverse of X. Equal to 1 divided by -12. Multiply by X, raise the power of -2, raise the power of -3 divided by 2. If we multiply the exponents -2 and -3 divided by 2, that will leave us with 1 divided by 1/2 multiplied by X raised to the power of 3. Finally, if we multiply the terms in the denominator, we will have 1 divided by negative X cubed divided by 2. And if we simplify the complex fraction, we will have F inverse prime of X equal to -2 divided by X cubed. This is going to be the derivative of the inverse function of F of X. And with that being said, the solution to this problem is going to be D. So, I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

Find the derivative of the inverse of the following functions. Express the result with x as the independent variable.
f(x) = x^-1/3

Key Concepts

Derivative

Inverse Functions

Chain Rule

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