Question

Find the 20 th derivative of f(x)=cos3x. Enclose arguments of functions in parentheses. For example, sin(2x). Enter your answer using exponents constants. For instance, write 3 6 instead of 729

f (20) (x)=_______

Answer 1

The 20th derivative of the function f(x) = cos(3x) is found by recognizing the periodicity of trigonometric derivatives and applying the chain rule. Since derivatives repeat every 4th step, the 20th derivative will be -3^(20)sin(3x).

Step-by-step explanation:

To find the 20th derivative of the function f(x) = cos(3x), we must recognize the periodic nature of the derivatives of trigonometric functions. The derivatives of cosine follow a cycle: cos → -sin → -cos → sin, and then the pattern repeats. Since the argument of the cosine function is 3x, each time we take a derivative, we will also introduce a factor of 3 (due to the chain rule).

Therefore, the sequence of derivatives at each step for cos(3x) looks like this: -3sin(3x), -9cos(3x), 27sin(3x), and so on. Every 4th derivative, we return to the original function multiplied by 81 (which is 3 raised to the power of 4). We want the 20th derivative, which is 5 cycles of this pattern (since 20 is 5 times 4). As 5 is an odd number, the 20th derivative will have the same sign as the first derivative in the cycle, which is negative. We will also multiply by 3 to the power of 20, because we introduce a factor of 3 each time we differentiate.

Therefore, the 20th derivative of f(x) = cos(3x) will be f(20)(x) = -3^(20)sin(3x).