Final answer:
The 65th derivative of y = cos(2x) is found by recognizing the cyclicality of derivatives of trigonometric functions. After 64 derivatives, we have 2^64 cos(2x), so the 65th derivative is -2^65 sin(2x), making option c) the correct answer.
Step-by-step explanation:
To find the 65th derivative of y = cos(2x), we need to recognize the pattern in the derivatives of trigonometric functions. The derivatives of sine and cosine functions are cyclical with a period of four.
Starting with cos(2x), its first derivative is -2 sin(2x), the second derivative is -4 cos(2x), the third derivative is 2(2 sin(2x)), and the fourth derivative returns to 4(2 cos(2x)), which is the original function multiplied by 2^2.
Now, to reach the 65th derivative, we divide 65 by 4, obtaining 16 with a remainder of 1. Since each cycle of four derivatives returns us to the original function multiplied by a power of four, the 64th derivative would be y = 2^64 cos(2x).
Therefore, the 65th derivative, being one derivative past this, will be the negative sine function multiplied by 2: -2(2^64) sin(2x) = -2^65 sin(2x).
So, the correct answer is option c) -2^65 sin(2x).
This pattern shows us that every time we take four more derivatives of a sine or cosine function, we multiply the original function by another power of four, and every set of four derivatives cycles through cosine to negative sine to negative cosine and finally to sine.