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

Find the derivative of 𝑒^(𝑥) sin 𝑥 using the product rule.

02:29

### Video Transcript

Find the derivative of 𝑒 to the power of 𝑥 multiplied by the sin of 𝑥 using the product rule.

We’re given the product of two functions, 𝑒 to the power of 𝑥 multiplied by the sin of 𝑥. And we need to find the derivative of this expression. And we’re told to do this by using the product rule.

Let’s start by recalling the product rule. The product rule tells us if we have two differentiable functions 𝑓 of 𝑥 and 𝑔 of 𝑥, then we can find the derivative of 𝑓 of 𝑥 multiplied by 𝑔 of 𝑥 with respect to 𝑥 as 𝑓 prime of 𝑥 multiplied by 𝑔 of 𝑥 plus 𝑔 prime of 𝑥 times 𝑓 of x. We’re told to use this on the function 𝑒 to the power of 𝑥 multiplied by the sin of 𝑥.

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, we’ll set 𝑓 of 𝑥 to be our first factor, 𝑒 to the power of 𝑥, and 𝑔 of 𝑥 to be our second factor, the sin of 𝑥. So now, 𝑒 to the power of 𝑥 times the sin of 𝑥 is actually 𝑓 of 𝑥 multiplied by 𝑔 of 𝑥. And this is what sets us up to use the product rule. But to use the product rule, we’re going to need to find expressions for 𝑓 prime of 𝑥 and 𝑔 prime of 𝑥.

Let’s start with finding 𝑓 prime of 𝑥. That’s the derivative of the exponential function 𝑒 to the power of 𝑥 with respect to 𝑥. And of course, we can evaluate this by recalling the derivative of the exponential function with respect to 𝑥 is just equal to 𝑒 to the power of 𝑥. So this means 𝑓 prime of 𝑥 is just 𝑒 to the power of 𝑥. We now need to find an expression for 𝑔 prime of 𝑥. That’s the derivative of the sin of 𝑥 with respect to 𝑥.

And this is a standard trigonometric derivative result which we should commit to memory. The derivative of the sin of 𝑥 with respect to 𝑥 is equal to the cos of 𝑥. So by using this, we’ve shown 𝑔 prime of 𝑥 is equal to the cos of 𝑥. We’re now ready to find an expression for the derivative of 𝑒 to the power of 𝑥 times the sin of 𝑥 with respect to 𝑥 by using the product rule.

We get this is equal to 𝑓 prime of 𝑥 times 𝑔 of 𝑥 plus 𝑔 prime of 𝑥 times 𝑓 of 𝑥. Now, we just need to substitute in our expressions for 𝑓 of 𝑥, 𝑔 of 𝑥, 𝑓 prime of 𝑥, and 𝑔 prime of 𝑥. We get 𝑒 to the power of 𝑥 times the sin of 𝑥 plus the cos of 𝑥 multiplied by 𝑒 to the power of 𝑥.

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