Video Transcript
Differentiate 𝑓 of 𝑥 is equal to 𝑒 to the power of 𝑥 times the tan of 𝑥.
We’re given a function 𝑓 of 𝑥 which is an exponential function multiplied by a trigonometric function. And we need to differentiate this. Since 𝑓 is a function of 𝑥, this will be the derivative of 𝑓 with respect to 𝑥. To do this, we need to notice that 𝑓 of 𝑥 is the product of two functions, and we know how to differentiate both of these functions. This means we can evaluate this derivative by using the product rule.
So let’s start by recalling the product rule. The product rule tells us that if we have two differentiable functions, 𝑢 of 𝑥 and 𝑣 of 𝑥, then the derivative of 𝑢 of 𝑥 multiplied by 𝑣 of 𝑥 with respect to 𝑥 is equal to 𝑢 prime of 𝑥 times 𝑣 of 𝑥 plus 𝑣 prime of 𝑥 times 𝑢 of 𝑥. In our case, 𝑓 of 𝑥 is the product of 𝑒 to the power of 𝑥 and the tan of 𝑥. So we’ll set our function 𝑢 of 𝑥 to be 𝑒 to the power of 𝑥 and 𝑣 of 𝑥 to be the tan of 𝑥.
Now, to use the product rule, we’re going to need to find expressions for 𝑢 prime of 𝑥 and 𝑣 prime of 𝑥. Let’s start with 𝑢 prime of 𝑥. That’s the derivative of 𝑒 to the power of 𝑥 with respect to 𝑥. And this is a standard derivative result we should know. The derivative of the exponential function 𝑒 to the power of 𝑥 with respect to 𝑥 is equal to itself, 𝑒 to the power of 𝑥. So 𝑢 prime of 𝑥 is just 𝑒 to the power of 𝑥.
Let’s now find an expression for 𝑣 prime of 𝑥. That’s the derivative of the tan of 𝑥 with respect to 𝑥. And there’s a few different ways we could evaluate this derivative. For example, we could write the tan of 𝑥 as the sin of 𝑥 divided by the cos of 𝑥 and then use the quotient rule. However, it’s easier to memorize this derivative. The derivative of the tan of 𝑥 with respect to 𝑥 is equal to the sec squared of 𝑥. So by applying this, we get 𝑣 prime of 𝑥 is equal to the sec squared of 𝑥. We’re now ready to find 𝑓 prime of 𝑥 by using the product rule. It’s the derivative of 𝑒 to the power of 𝑥 times the tan of 𝑥 with respect to 𝑥.
So by the product rule, this is equal to 𝑢 prime of 𝑥 times 𝑣 of 𝑥 plus 𝑣 prime of 𝑥 times 𝑢 of 𝑥. Substituting in 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 𝑒 to the power of 𝑥 times the tan of 𝑥 plus the sec squared of 𝑥 multiplied by 𝑒 to the power of 𝑥. And we could leave our answer like this. However, we’ll take out the shared factor of 𝑒 to the power of 𝑥. And doing this gives us our final answer of 𝑒 to the power of 𝑥 multiplied by the tan of 𝑥 plus the sec squared of 𝑥.
Therefore, by using the product rule on 𝑓 of 𝑥 is equal to 𝑒 to the power of 𝑥 times the tan of 𝑥, we were able to show that 𝑓 prime of 𝑥 is equal to 𝑒 to the power of 𝑥 multiplied by the tan of 𝑥 plus the sec squared of 𝑥.
