Question

How do you find the ${{27}^{th}}$ derivation of $\cos x$?

Answer 1

Hint: In this problem we have to find the ${{27}^{th}}$ derivation of $\cos x$.in this problem we will derivative the $\cos x$ in $27$ times . Then we will derive $\cos x$ then we will get $-\sin x$ we will do derivate only $4$ times only, because we have an ${{n}^{th}}$ derivation where $n$ is divisible by $4$ the derivative will be equal to $\cos x$. The closest multiple of $4$ to $27$ is $28$. The ${{28}^{th}}$ derivation of $\cos x$ is $\cos x$. Now we will make a list of four derivations.

Formula used:
1.$\dfrac{d}{dx}\left( \sin x \right)=\cos x$
2.$\dfrac{d}{dx}\left( \cos x \right)=-\sin x$

Complete step by step solution:
Given that, $\cos x$
Now we will derivative the above expression, then
$\dfrac{d}{dx}\left( \cos x \right)=-\sin x$
Now we will derivative the above expression, then
$\dfrac{d}{dx}\left( -\sin x \right)=-\cos x$
Now we will derivative the above expression, then
$\dfrac{d}{dx}\left( -\cos x \right)=\sin x$
Now we will derivative the above expression, then
$\dfrac{d}{dx}\left( \sin x \right)=\cos x$
Now we will stop differentiating $\cos x$ four times. We will return to $\cos x$.
Now we will make a list above derivatives, then
$1.$ first derivation is $-\sin x$
2. The second derivative is $-\cos x$
3. The third derivative is $\sin x$
4. Fourth derivative is $\cos x$
So, whenever we have an ${{n}^{th}}$ derivation where $n$ is divisible by $4$ the derivative will be equal to $\cos x$. The closest multiple of $4$to $27$ is $28$. The ${{28}^{th}}$ derivation of $\cos x$ is $\cos x$. Go one up in the list $\left( 3 \right)$

Hence the $27th$ derivative of $\cos x$ is $\sin x$.

Note: We can do the same method for $\sin x$ also. We will get the same repeating values. So, we will follow the above method for $\sin x$ also. Then we will get the final result.