### Video Transcript

Using differentials, find the approximate change in the surface area of a soap bubble when its radius increases from eight to 8.1 inches.

Now, we are actually told exactly how weβre going to solve this problem. Weβre told to use differentials. Given a function π¦ is equal to π of π₯, we call dπ¦ and dπ₯ differentials. And the relationship between them is given by dπ¦ equals π prime of π₯ dπ₯. So in this question, weβre going to need to begin by defining a function. And weβre going to define our function for surface area. Now, the surface area, letβs call that capital π΄, of a sphere with a radius π is given by four ππ squared. And so, weβre going to need to redefine a little.

We have π΄ as being some function of π. So in this case, our differentials are dπ΄ and dπ. And then, the relationship between them is dπ΄ equals π prime of π dπ. But what is dπ? Well, essentially, the differential of π is just a very small change in π. So in here, π changes from eight to 8.1 inches. So the change in π is the difference; itβs 0.1. Since the radius is increasing from eight, we can say that π is equal to eight. And dπ΄ is the small change in π΄ that weβre trying to find. And so, since we know dπ and we know π, itβs quite clear we need to find π prime of π. We defined π΄ to be equal to π of π. So thatβs four ππ squared. π prime of π is the first derivative of this function with respect to π.

Since four π is simply a constant, to differentiate four ππ squared, we multiply the entire term by the exponent and reduce that exponent by one. So we get two times four ππ to the power of one, which is simply eight ππ. In our formula, we said that π is equal to eight. So we need to find π prime of eight. Thatβs found by replacing π with eight. So we get eight π times eight, which is 64π. Then dπ΄ β remember, thatβs what weβre trying to find, itβs the change in the surface area β is the product of 64π and dπ which is 0.1.

We can now type this into our calculator. And we find that dπ΄ is equal to 20.10619 and so on, which, correct to three decimal places, is 20.106. And the approximate change in the surface area of the soap bubble is 20.106 square inches.