"Find the 50th derivative of y=cos⁡2x." was answered by our experts

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Calculus and Analysis
Differential Calculus
Derivatives

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2022-08-19

Find the 50th derivative of

y = \cos 2 x

Answer & Explanation

Catherine Kirby

Beginner2022-08-20Added 10 answers

Trigonometric derivatives cycle, as shown.
$y^{'} = - 2 \sin (2 x)$
$y^{″} = - 4 \cos (2 x)$
$y^{‴} = 8 \sin (2 x)$
$y^{⁗} = 16 \cos (2 x)$
Therefore, for the 50th derivative, the coefficient will be $2^{50}$ .
The remainder of 50 divided by 4 is 2, so we know it corresponds with y''.
50th derivative of $y = - 2^{50} \cos (2 x)$
Quadratic Formula.
$r = \frac{- - 4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$
$r = \frac{4 \pm \sqrt{16 - 4}}{2}$
$r = 2 \pm \sqrt{12}$
$r = 2 \pm 2 \sqrt{3}$
Answer:
Thus, 50th derivative of $y = - 2^{50} \cos (2 x)$

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