How to Find the Derivative of an Inverse at a Specific Point

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Finding the derivative of an inverse at a specific point is a process that begins by finding the inverse. Find the derivative of an inverse at a specific point with help from an experienced mathematics professional in this free video clip.

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Video Transcript

Hello, my name is Walter Unglaub. And this is how to find the derivative of an inverse at a specific point. So, if we start with the function Y as a function of X of one-eighth times X cubed. The first step is to find the inverse of this function. Which means that we want to solve for an equation of X, that is a function of Y. So, to do that, we multiply both sides by eight. So, we get X cubed is equal to eight-Y. Next, we take cube root of both sides, so were going to have X is equal to the cube root of eight-Y. We don't need a plus or a minus because this is an odd number power. And since eight is actually equal to two cubed. The two comes out of the cube root. So, we have two times Y raised to the power one-third. Next, we want to take the derivative of this inverse with respect to Y. So, I'm going to call my derivative X slash and it's going to be equal to DX divided DY or D, DY of my inverse function. Which is two-Y raised to the one-third power. So, X prime then becomes two divided three times one over Y raise to the power two-thirds. Because we subtract one from one-third and that gives us negative two-thirds. So, Y appears in the denominator. And finally, now that we have the derivative of the inverse to our original function, we can consider a point. So, if we consider originally the coordinate or the point, X,Y as two, one. When we take the inverse, of course, the roles are reversed. So, we have Y comma X and the two comma one becomes a one comma two. So, now we can plug in our Y values into the derivative of our inverse to calculate what the derivative is of the inverse at this particular point. And that will tell us information about the slope because the derivative is the slope. So, when Y is equal to one, X prime is equal to two-thirds times one over one raise to two-thirds. This is simply equal to one. Therefore, X prime is equal to two-thirds. And that is the slope or the derivative of the inverse of this particular function at the point one comma two. Where we have the order Y comma X. My name is Walter Unglaub, and this is how to find the derivative of an inverse at a specific point


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