### Video Transcript

Determine the derivative of π of π‘ equals π‘ sin of five ππ‘.

We have a function in π‘, which is actually the product of two functions. We could say that one of the functions is π‘. And the other is sin of five ππ‘. So how do we find the derivative of the product of two functions? For two functions π’ and π£ in π₯, the derivative of π’ times π£ is π’ times the derivative of π£ with respect to π₯ plus π£ times the derivative of π’ with respect to π₯.

Now, of course, our function is in terms of π‘. So we change this slightly to the derivative of π’π£ with respect to π‘. And this means we can let π’ be equal to π‘ because thatβs the first function in our equation. And we can let π£ be equal to sin of five ππ‘ because thatβs the second function of π‘.

We can see that weβre going to need to differentiate both of these with respect to π‘. If we differentiate π’ with respect to π‘, we simply get one. But differentiating π£ with respect to π‘ is a little bit trickier. We could use the chain rule. But we donβt need to. We can apply a general result. And that is if we differentiate sin of some constant of π‘, we get that constant multiplied by cos of that constant of π‘. So in this case, the derivative of sin of five ππ‘ is five π multiplied by cos of five ππ‘.

All thatβs left is to substitute this back into our equation for the product rule. π’ multiplied by ππ£ ππ‘ is π‘ multiplied by five π cos five ππ‘. And π£ multiplied by ππ’ ππ‘ is sin five ππ‘ multiplied by one, which simplifies to five ππ‘ multiplied by cos five ππ‘ plus sin of five ππ‘.