Question Video: Consecutive Derivatives of Sine Mathematics

Video Transcript

Find the 33rd derivative of 𝑓 of 𝑥 is equal to sin 𝑥.

In this question, we’re asked to determine the 33rd derivative of the sine function. We could do this by differentiating the sine function 33 times. However, it might be easier to try and find a pattern. Let’s start by finding an expression for 𝑓 prime of 𝑥. That’s the derivative of the sin of 𝑥 with respect to 𝑥. And we can recall this. The derivative of the sin of 𝑥 with respect to 𝑥 is the cos of 𝑥. This means to find the second derivative of 𝑓 of 𝑥, we need to differentiate the cosine function.

And we can also recall this. The derivative of the cos of 𝑥 with respect to 𝑥 is negative sin of 𝑥. So 𝑓 double prime of 𝑥, which is the derivative of the cos of 𝑥, is negative sin of 𝑥. We can keep going. To find the third derivative of 𝑓 of 𝑥, we need to differentiate negative sin of 𝑥. And we can also recall this. The derivative of negative sin of 𝑥 with respect to 𝑥 is negative cos of 𝑥. So 𝑓 triple prime of 𝑥 is equal to negative cos of 𝑥. Let’s do this one more time to find the fourth derivative of 𝑓 of 𝑥 with respect to 𝑥. We’re going to need to differentiate negative cos of 𝑥.

And we can recall the derivative of negative cos of 𝑥 with respect to 𝑥 is equal to the sin of 𝑥. This allows us to do two things. First, it allows us to find an expression for the fourth derivative of 𝑓 of 𝑥 with respect to 𝑥. We’ll write this in the alternate notation where the number in the brackets represents the number of times we’ve differentiated 𝑓 with respect to 𝑥. The fourth derivative of 𝑓 of 𝑥 is the sin of 𝑥. But the sin of 𝑥 was our original function. The fourth derivative of 𝑓 of 𝑥 is equal to 𝑓 of 𝑥. So every four times we differentiate 𝑓 of 𝑥 with respect to 𝑥, we end up back where we started with, the sin of 𝑥.

Therefore, if we differentiated this four more times with respect to 𝑥, we would end up back where we started. The eighth derivative of 𝑓 of 𝑥 with respect to 𝑥 is the sin of 𝑥. We want to find the 33rd derivative of 𝑓 of 𝑥. To do this, we note that 33 is not a multiple of four, but 32 is equal to four times eight. Therefore, we can get to the 32nd derivative of 𝑓 of 𝑥 by differentiating 𝑓 of 𝑥 in multiples of four. The 32nd derivative of 𝑓 of 𝑥 with respect to 𝑥 is the sin of 𝑥. We just need to differentiate this one more time to find the 33rd derivative of 𝑓 of 𝑥. We have the derivative of the sin of 𝑥 with respect to 𝑥 is the cos of 𝑥. Therefore, we were able to show the 33rd derivative of 𝑓 of 𝑥 is equal to sin of 𝑥 is the cos of 𝑥.