find the 61th derivative of the function ( f(x)=cos (x) ).\nthe answer is function

Explanation:

Step1: Find the pattern of derivatives

The first - derivative: (f^{\prime}(x)=-\sin(x)) The second - derivative: (f^{\prime\prime}(x)=-\cos(x)) The third - derivative: (f^{\prime\prime\prime}(x)=\sin(x)) The fourth - derivative: (f^{(4)}(x)=\cos(x)) The pattern repeats every (4) derivatives.

Step2: Divide the order of the derivative by (4)

We want to find the (61)th derivative. Divide (61) by (4): (61 = 4\times15+1)

Step3: Determine the (61)th derivative

Since the remainder is (1) when (61) is divided by (4), the (61)th derivative has the same form as the first - derivative.

Answer:

(-\sin(x))