# Question Video: Finding the Derivative of a Function Involving Trigonometric and Exponential Functions Using the Product Rule Mathematics • Higher Education

Find the derivative of π^(π₯) sin π₯ using the product rule.

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### Video Transcript

Find the derivative of π to the power of π₯ multiplied by the sin of π₯ using the product rule.

Weβre given the product of two functions, π to the power of π₯ multiplied by the sin of π₯. And we need to find the derivative of this expression. And weβre told to do this by using the product rule.

Letβs start by recalling the product rule. The product rule tells us if we have two differentiable functions π of π₯ and π of π₯, then we can find the derivative of π of π₯ multiplied by π of π₯ with respect to π₯ as π prime of π₯ multiplied by π of π₯ plus π prime of π₯ times π of x. Weβre told to use this on the function π to the power of π₯ multiplied by the sin of π₯.

And we can see this is in fact the product of two functions which are differentiable. This means we can use the product rule. To start, weβll set π of π₯ to be our first factor, π to the power of π₯, and π of π₯ to be our second factor, the sin of π₯. So now, π to the power of π₯ times the sin of π₯ is actually π of π₯ multiplied by π of π₯. And this is what sets us up to use the product rule. But to use the product rule, weβre going to need to find expressions for π prime of π₯ and π prime of π₯.

Letβs start with finding π prime of π₯. Thatβs the derivative of the exponential function π to the power of π₯ with respect to π₯. And of course, we can evaluate this by recalling the derivative of the exponential function with respect to π₯ is just equal to π to the power of π₯. So this means π prime of π₯ is just π to the power of π₯. We now need to find an expression for π prime of π₯. Thatβs the derivative of the sin of π₯ with respect to π₯.

And this is a standard trigonometric derivative result which we should commit to memory. The derivative of the sin of π₯ with respect to π₯ is equal to the cos of π₯. So by using this, weβve shown π prime of π₯ is equal to the cos of π₯. Weβre now ready to find an expression for the derivative of π to the power of π₯ times the sin of π₯ with respect to π₯ by using the product rule.

We get this is equal to π prime of π₯ times π of π₯ plus π prime of π₯ times π of π₯. Now, we just need to substitute in our expressions for π of π₯, π of π₯, π prime of π₯, and π prime of π₯. We get π to the power of π₯ times the sin of π₯ plus the cos of π₯ multiplied by π to the power of π₯.

And we could leave our answer like this. However, we can also simplify by taking out the common factor of π to the power of π₯. And this gives us our final answer: π to the power of π₯ times the sin of π₯ plus the cos of π₯. Therefore, we were able to find the derivative of π to the power of π₯ multiplied by the sin of π₯ with respect to π₯ by using the product rule. We got π to the power of π₯ times the sin of π₯ plus the cos of π₯.