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.
