Question

How do you find the 108th derivative of $y=\cos \left( x \right)$?

Answer 1

Now we will differentiate the function again and again till we see a pattern in the differential obtained. Now with the help of this pattern we can see that every $4{{n}^{th}}$ derivative of the function the cycle repeats as the fourth derivative is $\cos x$ . Hence we can easily find the ${{108}^{th}}$ derivative of the function.

Complete step by step solution:
Now consider the function $y=\cos x$ . Now we know that the derivative of $\cos x$ is $-\sin x$ .
Hence we can say that the first derivative of the function $y=\cos \left( x \right)$ is $-\sin x$ .
Now the derivative of $\sin x$ is nothing but $\cos x$ .
Hence we can say that the second derivative of the function $y=\cos \left( x \right)$ is $-\cos x$ .
Now again differentiating the function we get the third derivative of the function $y=\cos \left( x \right)$ as $\sin x$ .
And finally again differentiating the function we get the fourth derivative of the function $y=\cos \left( x \right)$ as $\cos x$ . Hence we can see that the derivatives of $\cos x$ repeats a cycle after every fourth derivative.
Hence after differentiating the function 4n times we will still get $\cos x$
Now we want to differentiate the function 108 times.
Now we know that 108 is nothing but $27\times 4$ .
Hence we want to differentiate the function 4n times where n = 27.

Now we know that on differentiating the function 4 times we get cosx. Hence the ${{108}^{th}}$ derivative of $\cos x$ is nothing but $\cos x$.

Note: Now note that if the ${{n}^{th}}$ derivative is not divisible by 4 then we write n as 4k + r. where r is the remainder obtained after dividing by 4. Hence the ${{n}^{th}}$ derivative is same as ${{r}^{th}}$ derivative and we can calculate ${{r}^{th}}$ derivative as r is less than 4.