Question

What is the 76th derivative of the function f(x) = 3sin(2x)? a) 2^76 * 3sin(2x) b) 2^76 * 3cos(2x) c) 2^76 * (-3)sin(2x) d) 2^76 * (-3)cos(2x)

Answer 1

The 76th derivative of f(x) = 3sin(2x) is 2^76 * 3sin(2x), we account for the chain rule with the inner derivative of 2x, so the correct option is a.

Step-by-step explanation:

The 76th derivative of the function f(x) = 3sin(2x) can be found by recognizing that the derivatives of sine and cosine functions are periodic.

They repeat every four derivatives. The derivatives of sine are as follows: the first derivative is cosine, the second is negative sine, the third is negative cosine, and the fourth is sine again.

Every time we take the derivative of sine or cosine with respect to x, we also multiply by the derivative of the inner function due to the chain rule.

Here, the inner function is 2x, which has a derivative of 2. Thus, every time we differentiate, we introduce an additional factor of 2.

Now, since 76 is a multiple of 4, the 76th derivative of sin(2x) will give us sin(2x) again, but multiplied by the factor 2^76.

Since the 76th derivative maintains the sine function without a phase shift or sign change, the correct answer is 2^76 * 3sin(2x).

So, the correct option is a.