# Video: Finding the nth Derivative of an Expressions by Finding the First Few Derivatives and Observing the Pattern That Occurs

Find d⁵¹/d𝑥⁵¹ (sin 𝑥) by finding the first few derivatives and observing the pattern that occurs.

02:49

### Video Transcript

Find the 51st derivative of sin of 𝑥 with respect to 𝑥 by finding the first few derivatives and observing the pattern that occurs.

Let’s begin then by finding the first few derivatives of sin of 𝑥 with respect to 𝑥. We can quote the standard result that d by d𝑥 of sin of 𝑥 is cos of 𝑥. This means to find the second derivative of sin of 𝑥, we need to differentiate cos of 𝑥 with respect to 𝑥.

Here, we can quote another standard result. The derivative of cos of 𝑥 with respect to 𝑥 is negative sin of 𝑥. So, the second derivative of sin of 𝑥 with respect to 𝑥 is negative sin of 𝑥. Similarly, the third derivative is going to be found by differentiating negative sin 𝑥 with respect to 𝑥. And we can use the constant multiple rule here to take the constant of negative one outside the derivative and concentrate on differentiating sin of 𝑥.

We’ve already seen that the derivative of sin of 𝑥 with respect to 𝑥 is cos of 𝑥. So, this means that the third derivative of sin of 𝑥 with respect to 𝑥 is negative cos of 𝑥. The fourth derivative of sin of 𝑥 is going to be the derivative of negative cos of 𝑥 with respect to 𝑥. Once again, we’ll use the constant multiple rule here and take the constant of negative one outside the derivative and concentrate on differentiating cos of 𝑥, which we now know to be negative sin of 𝑥.

So, the fourth derivative is negative negative sin of 𝑥, which is positive sin of 𝑥. And we don’t actually need to do anymore. We can see that we have a cycle. The fifth derivative of sin of 𝑥 is going to be cos of 𝑥. And the sixth derivative will go back to negative sin of 𝑥, and so on. So, what’s the general rule?

Well, we can say that for integer values of 𝑘, the four 𝑘th derivative of sin of 𝑥 is sin of 𝑥. The four 𝑘th plus oneth derivative of sin of 𝑥 is cos of 𝑥. The four 𝑘 plus twoth derivative of sin of 𝑥 is negative sin of 𝑥. And the four 𝑘 plus threeth derivative of sin of 𝑥 is negative cos of 𝑥.

We’re trying to find the 51st derivative. And we can write 51 as four times 12 plus three. So, this means that the 51st derivative of sin of 𝑥 will be the same as the four 𝑘 plus threeth derivative. It’s negative cos of 𝑥.

It’s useful to know that since the derivatives of sin and cos are so closely related, we can also derive a general formula for the 𝑛th derivative of cos of 𝑥. The four 𝑘th derivative of cos of 𝑥 is cos of 𝑥. The four 𝑘 plus oneth derivative of cos of 𝑥 is negative sin of 𝑥. The four 𝑘 plus twoth derivative of cos of 𝑥 is negative cos of 𝑥. And the four 𝑘 plus threeth derivative of cos of 𝑥 is sin of 𝑥.