Video Transcript
Differentiate 𝑓 of 𝑥 is equal to 𝑒 to the power of 𝑥 times the sin of 𝑥.
The question wants us to differentiate our function 𝑓 of 𝑥. And we can see that our function is the product of two functions. It’s the product of 𝑒 to the 𝑥 and the sin of 𝑥. And we know how to differentiate the product of two functions by using the product rule. The product rule tells us if we have a function 𝑓 of 𝑥 which is the product of two functions 𝑢 of 𝑥 and 𝑣 of 𝑥, then 𝑓 prime of 𝑥 is equal to 𝑣 of 𝑥 times 𝑢 prime of 𝑥 plus 𝑢 of 𝑥 times 𝑣 prime of 𝑥.
And 𝑓 of 𝑥 is the product of two functions. So, we’ll set 𝑢 of 𝑥 to be 𝑒 to the power of 𝑥 and 𝑣 of 𝑥 to be the sin of 𝑥. To use the product rule, we need to find expressions for 𝑢 prime of 𝑥 and 𝑣 prime of 𝑥. Let’s start with 𝑢 prime of 𝑥. We see that 𝑢 prime of 𝑥 will be the derivative of 𝑒 to the power of 𝑥 with respect to 𝑥. And we know that this differentiates to give itself. So, 𝑢 prime of 𝑥 is also equal to 𝑒 to the power of 𝑥.
Next, let’s find an expression for 𝑣 prime of 𝑥. That’s the derivative of the sin of 𝑥 with respect to 𝑥. And this is a standard trigonometric derivative result that we should commit to memory. The derivative of the sin of 𝑥 with respect to 𝑥 is equal to the cos of 𝑥. So, 𝑣 prime of 𝑥 is equal to the cos of 𝑥.
We’re now ready to find an expression for our derivative function 𝑓 prime of 𝑥. By the product rule, it’s equal to 𝑣 of 𝑥 times 𝑢 prime of 𝑥 plus 𝑢 of 𝑥 times 𝑣 prime of 𝑥. Substituting our expressions for 𝑢 of 𝑥, 𝑣 of 𝑥, 𝑢 prime of 𝑥, and 𝑣 prime of 𝑥, we get that 𝑓 prime of 𝑥 is equal to the sin of 𝑥 times 𝑒 to the power of 𝑥 plus 𝑒 to the power of 𝑥 times the cos of 𝑥.
We could leave our answer like this; however, we’ll take out the shared factor of 𝑒 to the power of 𝑥. And this gives us 𝑒 to the power of 𝑥 times the sin of 𝑥 plus the cos of 𝑥. And this is our final answer. Therefore, we’ve shown if 𝑓 of 𝑥 is equal to 𝑒 to the power of 𝑥 times the sin of 𝑥, then 𝑓 prime of 𝑥 is equal to 𝑒 to the power of 𝑥 times the sin of 𝑥 plus the cos of 𝑥.
