Scelta multipla
Differentiate the function: . Which of the following is the correct derivative ?
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3. Techniques of Differentiation
Derivatives of Trig Functions
Scelta multipla
Find the indicated derivative.
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Understand that the problem requires finding the 19th derivative of the function \( f(x) = \cos x \).
Recall that the derivatives of \( \cos x \) follow a cyclical pattern: \( f'(x) = -\sin x \), \( f''(x) = -\cos x \), \( f'''(x) = \sin x \), and \( f^{(4)}(x) = \cos x \). This cycle repeats every four derivatives.
Determine the position of the 19th derivative within this cycle. Since the cycle repeats every 4 derivatives, calculate \( 19 \mod 4 \) to find the position within the cycle.
Calculate \( 19 \mod 4 \), which equals 3. This means the 19th derivative corresponds to the third derivative in the cycle.
Identify the third derivative in the cycle, which is \( \sin x \). Therefore, \( f^{(19)}(x) = \sin x \).
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