### 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.

Letβs 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₯.

Weβve 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, weβll 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 donβt 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, whatβs the general
rule?

Well, we can say that for
integer values of π, the four πth derivative of sin of π₯ is sin of π₯. The four πth 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₯.

Weβre 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. Itβs negative cos of π₯.

Itβs useful to know that since
the derivatives of sin and cos are so closely related, we can also derive a
general formula for the πth derivative of cos of π₯. The four πth 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₯.