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

Find dβ΅ΒΉ/dπ₯β΅ΒΉ (sin π₯) by finding the first few derivatives and observing the pattern that occurs.

02:49

### Video Transcript

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

Letβs begin then by finding the first few derivatives of sin of π₯ with respect to π₯. We can quote the standard result that d by dπ₯ of sin of π₯ is cos of π₯. This means to find the second derivative of sin of π₯, we need to differentiate cos of π₯ with respect to π₯.

Here, we can quote another standard result. The derivative of cos of π₯ with respect to π₯ is negative sin of π₯. So, the second derivative of sin of π₯ with respect to π₯ is negative sin of π₯. Similarly, the third derivative is going to be found by differentiating negative sin π₯ with respect to π₯. 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 π₯.

Weβve already seen that the derivative of sin of π₯ with respect to π₯ is cos of π₯. So, this means that the third derivative of sin of π₯ with respect to π₯ is negative cos of π₯. The fourth derivative of sin of π₯ is going to be the derivative of negative cos of π₯ with respect to π₯. Once again, weβll use the constant multiple rule here and take the constant of negative one outside the derivative and concentrate on differentiating cos of π₯, which we now know to be negative sin of π₯.

So, the fourth derivative is negative negative sin of π₯, which is positive sin of π₯. And we donβt actually need to do anymore. We can see that we have a cycle. The fifth derivative of sin of π₯ is going to be cos of π₯. And the sixth derivative will go back to negative sin of π₯, and so on. So, whatβs the general rule?

Well, we can say that for integer values of π, the four πth derivative of sin of π₯ is sin of π₯. The four πth plus oneth derivative of sin of π₯ is cos of π₯. The four π plus twoth derivative of sin of π₯ is negative sin of π₯. And the four π plus threeth derivative of sin of π₯ is negative cos of π₯.

Weβ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 π₯ will be the same as the four π plus threeth derivative. Itβs negative cos of π₯.

Itβ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 πth derivative of cos of π₯. The four πth derivative of cos of π₯ is cos of π₯. The four π plus oneth derivative of cos of π₯ is negative sin of π₯. The four π plus twoth derivative of cos of π₯ is negative cos of π₯. And the four π plus threeth derivative of cos of π₯ is sin of π₯.