Question

exercises 3.5 derivatives of trigonometric
score: 10/12 answered: 11/12
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question 12
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find the 27th derivative of the function f(x)=cos(x).
the answer is function
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Explanation:

Step1: Find the pattern of derivatives

The first - derivative of \(y = \cos(x)\) is \(y'=-\sin(x)\). The second - derivative \(y''=-\cos(x)\). The third - derivative \(y'''=\sin(x)\). The fourth - derivative \(y^{(4)}=\cos(x)\). The pattern of derivatives of \(y = \cos(x)\) repeats every \(n = 4\).

Step2: Divide the order of the derivative by 4

We want to find the 27th derivative. Divide 27 by 4: \(27=4\times6 + 3\).

Step3: Determine the 27th derivative

Since the pattern repeats every 4 derivatives, and \(27 = 4\times6+3\), the 27th derivative of \(y=\cos(x)\) has the same form as the 3rd derivative.

So, the 27th derivative of \(f(x)=\cos(x)\) is \(\sin(x)\).

Answer:

$\sin(x)$