Now that you know how to find the derivatives of some trigonometric functions like sine and cosine, you may also be asked to find some higher order derivatives of these functions, like, say, the 37th derivative of sine. Now we know that higher order derivatives are just repeated differentiation, but you probably don't want to have to take the derivative here 37 times. And luckily, you don't have to because here I'm going to show you a shortcut specifically for finding the higher order derivatives of our sine and cosine functions much more quickly and easily. So let's go ahead and dive right in here. Now here we're asked to find the 37th derivative of sine.
We, of course, don't want to have to take the derivative 37 times, but let's go ahead and take the first couple of derivatives and see what happens. Now if we take the first derivative d/dx of the sine of x, that gives us the cosine of x. If I take the derivative again, finding my second derivative, that's going to give me the negative sine of x. Now if I take the derivative again here, differentiating this negative sine in order to get my 3rd derivative, that's going to give me the negative cosine of x. Now, if I take one more derivative here, differentiating this negative cosine of x to get that 4th derivative, this is going to give me negative negative sine of x, which these two negatives just give me a positive sine of x, which is the exact same function that I started with, the sine of x.
This is actually going to happen for every 4th derivative. For every multiple of 4, the 8th derivative, the 12th derivative, the 24th derivative, all of these are just going to be our original function, sine. So because of that, in order to find a higher order derivative, we can just divide n by 4 and then take derivatives based on whatever our remainder is. So here when finding the 37th derivative, our n value is 37. If I take 37 and divide it by 4, I know that 4 times 9 is 36. So when I divide 37 by 4, this is going to give me 9 with a remainder of 1 to get up to that 37. So because of that, this 37th derivative is the exact same as just the first derivative of the sine of x, taking my derivatives here based on my remainder of 1. Now the derivative of sine, we know, is just equal to the cosine, and that's our final answer. This is the 37th derivative of sine. That's all there is to it.
Now let's take a look at our next example where we're asked to find the 58th derivative of the cosine of x. Now higher order derivatives of cosine work the same exact way. Every 4th derivative of cosine will also be cosine. So we need to do the same thing here. Just divide n by 4.
I have 58 for that n value here if I divide it by 4. Thinking about multiplying 4 by different numbers, I know that 4 times 14 is going to give me 56. So taking 58 and dividing it by 4 is going to give me 14 with a remainder of 2 to get up to that 58. So now we want to take derivatives based on that remainder of 2. So this means that the 58th derivative of cosine is the exact same as just the second derivative of cosine, so we only need to take the derivative twice.
Now we know that the first derivative d/dx of the cosine of x is equal to the negative sine of x. Taking the derivative one more time here to find our second-order derivative of the cosine of x, differentiating this negative sine gives me negative cosine of x, and this is my final answer. This is the 58th derivative of cosine. So, whenever you're faced with higher order derivatives of sine or cosine, just divide that n value by 4. Take derivatives based only on the remainder. Don't worry about differentiating 58 or 37 times. Let's get some more practice with this. I'll see you in the next video.