Question

find the 61th derivative of the function ( f(x)=cos (x) ).
the 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:

\(-\sin(x)\)