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

Find d⁡¹/dπ‘₯⁡¹ (sin π‘₯) by finding the first few derivatives and observing the pattern that occurs.

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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.

Let’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‘₯.

We’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, we’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 don’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, what’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‘₯.

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 π‘₯ will be the same as the four π‘˜ plus threeth derivative. It’s negative cos of π‘₯.

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 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‘₯.

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