# How do you find #f^37x# given #f(x)=cos3x#?

For problems of this kind, they want you to find a pattern for the derivatives.

By the chain rule, the first derivative is:

The second derivative is :

The third derivative is:

The fourth derivative is:

The fifth derivative is:

As you can imagine, this is a massive number. Therefore, we'll have to keep it in the form that it is above.

Hopefully this helps!

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To find ( f^{37}(x) ) given ( f(x) = \cos(3x) ), use the fact that ( f^{n}(x) ) represents the nth derivative of ( f(x) ). Since ( f(x) = \cos(3x) ), its derivatives follow a pattern where the derivative of ( \cos(ax) ) with respect to ( x ) is ( -a\sin(ax) ). Hence, the 37th derivative of ( \cos(3x) ) is ( (-3)^{37}\sin(3x) ). Therefore, ( f^{37}(x) = (-3)^{37}\sin(3x) ).

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