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Find the 10th derivative of y = cos 3x.

User Brian Mego
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Final answer:

The 10th derivative of y = cos 3x is -59049 sin(3x), as the derivatives of cosine follow a repeating cyclical pattern every four derivatives, accounting for the chain rule.

Step-by-step explanation:

When asked to find the 10th derivative of y = cos 3x, we are dealing with a straightforward calculus problem. The derivatives of cosine functions follow a cyclical pattern that repeats every four derivatives. Since the function involves cos 3x, multiplying factors of 3 will appear in the derivatives due to the chain rule.

The original function y can be represented as y = cos(3x).

Taking the first derivative y' gives us -3 sin(3x), and the second derivative y'' would be -9 cos(3x).

Continuing this process, we would find that every even derivative is a multiple of the original function cos(3x) or -cos(3x), and every odd derivative is a multiple of sin(3x) or -sin(3x).

On the 4th, 8th, and 12th derivatives (and so on every 4th derivative), we return to the original function cos(3x).

As such, the 8th derivative would be 9^4 cos(3x) and the 10th derivative, following two more steps in the cycle, becomes -9^5 sin(3x).

Therefore, the 10th derivative of y = cos 3x is -9^5 sin(3x), or explicitly written, -59049 sin(3x).

User DotNetDublin
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