### Video Transcript

If π¦ equals sin of eight π₯ squared minus four, find dπ¦ by dπ₯.

Well, this is a function of a function. We have eight π₯ squared minus four as the inner function and sine as the outer function. And when we want to differentiate a function of a function, we use the chain rule. The chain rule says that if π¦ equals π of π’ and π’ equals π of π₯ then dπ¦ by dπ₯ equals dπ¦ by dπ’ multiplied by dπ’ by dπ₯. So for our question, π’ equals eight π₯ squared minus four. And so, π¦ equals sin of π’.

Now, our formula for the chain rule requires us to have dπ’ by dπ₯ and dπ¦ by dπ’. Letβs start by differentiating π’ with respect to π₯. We remember the power rule of differentiating. And so, dπ’ by dπ₯ equals 16π₯. This is because we multiplied eight, the coefficient, by the power which is two to get 16. And then, we took one away from the power. And four is a constant which differentiates to zero.

Now, letβs find dπ¦ by dπ’. We recall that the derivative of sin of π₯ is cos of π₯. And so, dπ¦ by dπ’ equals cos of π’. Okay, so now, letβs apply the formula for the chain rule. We have that dπ¦ by dπ₯ equals dπ¦ by dπ’, which is cos of π’, multiplied by dπ’ by dπ₯, which is 16π₯. Remember that we defined π’ to be eight π₯ squared minus four. So, we can replace π’ in our answer to give us that dπ¦ by dπ₯ equals cos of eight π₯ squared minus four multiplied by 16π₯, which we usually write the other way round as 16π₯ cos of eight π₯ squared minus four. So, that is our final answer.