### Video Transcript

If π is the volume of a cube with edge length π₯ and the cube expands as time passes, give an expression for ππ ππ‘.

So first of all, Iβm gonna take a look at what weβve actually given in the question. We know that π is the volume and π₯ is the edge length. So therefore, we can say that π is equal to π₯ cubed. So the volume is equal to π₯ cubed. And thatβs because if we tried to work out the volume of a cube, all we do is we actually cube one of the lengths of the sides. Okay, great, so weβve got our first little expression there.

So now, the next step would actually to be work out ππ ππ₯. So we actually gonna differentiate the value that we had earlier which was that π is equal to π₯ cubed. So ππ ππ₯ can be equal to three π₯ squared. And weβll reach that because if weβre multiplying the exponent by the coefficient, thatβs three multiplied by one which gives us three. And then, all weβve done is weβve actually reduced the exponent by one because the three minus one which gives us two. So we get three π₯ squared.

So now as weβre actually looking to find ππ ππ‘, weβre now gonna use the chain rule, which states that ππ¦ ππ₯ is equal to ππ¦ ππ‘ multiplied by ππ‘ ππ₯. Well, in our case, weβre gonna get ππ ππ‘ because this is like ππ¦ ππ₯, which is going to be equal to ππ ππ₯ multiplied by ππ₯ ππ‘. And we get that because if we take a look, our ππ₯ terms will actually cancel out because we will have a ππ₯ on the numerator, ππ₯ on the denominator. So then, weβll gonna get ππ ππ‘.

Okay, we already know ππ ππ₯ from our previous step. However, we donβt know ππ₯ ππ‘. So therefore, ππ ππ‘ what it actually means in practice, which is the change in the volume of the cube over time, is equal to three π₯ squared ππ₯ ππ‘.