# Video: Using Differentials to Find the Change in Surface Area of a Sphere

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

02:26

### 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 weβre going to solve this problem. Weβre 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, weβre going to need to begin by defining a function. And weβre going to define our function for surface area. Now, the surface area, letβs call that capital π΄, of a sphere with a radius π is given by four ππ squared. And so, weβre 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; itβs 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 weβre trying to find. And so, since we know dπ and we know π, itβs quite clear we need to find π prime of π. We defined π΄ to be equal to π of π. So thatβs four ππ 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 ππ squared, we multiply the entire term by the exponent and reduce that exponent by one. So we get two times four ππ to the power of one, which is simply eight ππ. In our formula, we said that π is equal to eight. So we need to find π prime of eight. Thatβs found by replacing π with eight. So we get eight π times eight, which is 64π. Then dπ΄ β remember, thatβs what weβre trying to find, itβs 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.