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Clifland

Answered question

2021-10-17

Find the second order derivatives of the function

\cos^{3} x

grbavit

Skilled2021-10-18Added 109 answers

Step 1
By the formula:

\cos (3 x) = 4 \cos^{3} (x) - 3 \cos (x)

4 \cos^{3} (x) = \cos (3 x) + 3 \cos (x)

\cos 3 (x) = \frac{1}{4} (\cos (3 x)) + \frac{3}{4} (\cos (x))

Let

f (x) = \cos^{3} (x)

= \frac{1}{4} \cos (3 x) + \frac{3}{4} \cos (x)

Step 2
Then

f^{'} (x) = \frac{d}{d x} [\frac{1}{4} \cos (3 x) + \frac{3}{4} \cos (x)]

= \frac{1}{4} (- 3 \sin (3 x)) + \frac{3}{4} (- \sin (x))

= \frac{- 3}{4} \sin (3 x) - \frac{3}{4} \sin (x)

Then

f^{″} (x) = \frac{d}{d x} f^{'} (x)

= \frac{d}{d x} [\frac{- 3}{4} \sin (3 x) - \frac{3}{4} \sin (x)]

= \frac{- 3}{4} (3 \cos (3 x) - \frac{3}{4} (\cos (x))

= - \frac{9}{4} \cos (3 x) - \frac{3}{4} \cos (x)

Therefore

(\cos^{3} x)^{″} = - \frac{9}{4} \cos (3 x) - \frac{3}{4} \cos (x)

Jeffrey Jordon

Expert2022-11-14Added 2605 answers

Answer is given below (on video)

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