Video Transcript
Differentiate 𝑓 of 𝑥 is equal to 𝑒 to the power of 𝑥 times the sec of 𝑥.
The question wants us to differentiate 𝑓 of 𝑥 which is the product of two functions. To differentiate the product of two functions, we recall the product rule for differentiation, which tells us if 𝑢 and 𝑣 are both functions of 𝑥, then the derivative of 𝑢 times 𝑣 with respect to 𝑥 is equal to 𝑢 prime 𝑣 plus 𝑢𝑣 prime. So to differentiate our function 𝑓 of 𝑥, we’ll set 𝑢 equal to 𝑒 to the power of 𝑥 and 𝑣 equal to the sec of 𝑥. Then, to find our function 𝑢 prime, we need to differentiate 𝑒 to the power of 𝑥. And we recall the derivative of 𝑒 to the power of 𝑥 with respect to 𝑥 is just equal to itself. So 𝑢 prime is equal to 𝑒 to the power of 𝑥.
Next, to find 𝑣 prime, we need to differentiate the sec of 𝑥. We recall a standard rule for trigonometric derivatives. The derivative of the sec of 𝑥 with respect to 𝑥 is equal to the sec of 𝑥 times the tan of 𝑥. You could prove this by writing the sec of 𝑥 as one divided by the cos of 𝑥 and then using the quotient rule for differentiation. However, it’s useful to commit this to memory. Using this, we see that 𝑣 prime is equal to the sec of 𝑥 times the tan of 𝑥.
We’re now ready to apply the product rule to find the derivative of 𝑓 of 𝑥. It’s equal to 𝑢 prime 𝑣 plus 𝑢𝑣 prime. Substituting the functions we found for 𝑢, 𝑣, 𝑢 prime, and 𝑣 prime, we have that 𝑓 prime of 𝑥 is equal to 𝑒 to the power of 𝑥 times the sec of 𝑥 plus 𝑒 to the power of 𝑥 times the sec of 𝑥 times the tan of 𝑥. And we could leave our answer like this. However, we can simplify our answer by noticing that both terms share a factor of 𝑒 to the 𝑥 and both terms share a factor of the sec of 𝑥.
So we take out our factor of 𝑒 to the power of 𝑥 times the sec of 𝑥. We then see that we need to multiply this by one plus the tan of 𝑥. And this gives us our final answer. If 𝑓 of 𝑥 is equal to 𝑒 to the power of 𝑥 times the sec of 𝑥, then the derivative of 𝑓 of 𝑥, 𝑓 prime of 𝑥, is equal to 𝑒 to the power of 𝑥 times the sec of 𝑥 multiplied by one plus the tan of 𝑥.
