### Video Transcript

Find the first derivative of the function π¦ is equal to π to the power of π₯ times the sin of seven π₯.

Weβre told that π¦ is equal to π to the power of π₯ times the sin of seven π₯. We need to find the first derivative of this expression for π¦. And since π¦ is a function of π₯, this means we need to find the derivative of π¦ with respect to π₯. We can do this by noticing that π¦ is the product of two functions. Itβs π to the power of π₯ multiplied by the sin of seven π₯. And in fact, itβs the product of two differentiable functions. We know how to find their derivatives.

So we can differentiate this by using the product rule. We recall the product rule tells us for differentiable functions π of π₯ and π of π₯, the derivative of π of π₯ multiplied by π of π₯ with respect to π₯ is equal to π prime of π₯ times π of π₯ plus π prime of π₯ times π of π₯. In our case, we want to find the first derivative of π¦ with respect to π₯. Thatβs the derivative of π to the power of π₯ multiplied by the sin of seven π₯ with respect to π₯.

So we can do this by using the product rule. We need to set π of π₯ equal to π to the power of π₯ and π of π₯ to be equal to the sin of seven π₯. This means weβre going to need to find expressions for π prime of π₯ and π prime of π₯. Letβs start with π prime of π₯. Thatβs the derivative of π to the power of π₯ with respect to π₯. And we can do this by recalling the derivative of the exponential function π to the power of π₯ is just equal to itself, π to the power of π₯.

So π prime of π₯ is just equal to π to the power of π₯. We now need to find an expression for π prime of π₯. Thatβs the derivative of the sin of seven π₯ with respect to π₯. And to do this, we need to recall one of our standard trigonometric derivative results. For any real constant π, the derivative of the sin of ππ₯ with respect to π₯ is equal to π times the cos of ππ₯. In this case, the value of π is equal to seven. So we get π prime of π₯ is equal to seven times the cos of seven π₯.

Weβre now ready to find the first derivative of π¦ with respect to π₯ by using the product rule. Itβs equal to π prime of π₯ times π of π₯ plus π prime of π₯ times π of π₯. Substituting in our expressions for π of π₯, π of π₯, π prime of π₯, and π prime of π₯, we get that π¦ prime is equal to π to the power of π₯ times the sin of seven π₯ plus seven times the cos of seven π₯ multiplied by π to the power of π₯.

And we could leave our answer like this or we could take out the common factor of π to the power of π₯. However, instead, weβll leave this as two terms, and weβll write π to the power of π₯ at the front of our second term. We do this to avoid confusion of the cos of seven π₯ all multiplied by π to the power of π₯ with the cos of seven π₯ times π to the power of π₯. And this gives us our final answer.

Therefore, given π¦ is equal to π to the power of π₯ times the sin of seven π₯, we were able to find the first derivative of π¦ with respect to π₯ by using the product rule. We got that π¦ prime was equal to π to the power of π₯ times the sin of seven π₯ plus seven π to the power of π₯ multiplied by the cos of seven π₯.