"Find the derivatives of the functions y=(3+cos33x)−13" was answered by our experts

hvacwk

Answered question

2021-12-11

Find the derivatives of the functions

y = {(3 + \cos^{3} 3 x)}^{- \frac{1}{3}}

Answer & Explanation

otoplilp1

Beginner2021-12-12Added 41 answers

Step 1
Given that,

y = {(3 + \cos^{3} 3 x)}^{- \frac{1}{3}}

Step 2
On differentiating y with respect to x,

\frac{d y}{d x} = \frac{d}{d x} [{(3 + \cos^{3} 3 x)}^{- \frac{1}{3}}]

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

= - \frac{1}{3} {(3 + \cos^{3} 3 x)}^{- \frac{4}{3}} (0 + 3 \cos^{2} 3 x \frac{d}{d x} (\cos 3 x))

= - \frac{1}{3} {(3 + \cos^{3} 3 x)}^{- \frac{4}{3}} (3 \cos^{2} 3 x \sin 3 x (3))

= 3 \cos^{2} 3 x \sin 3 x {(3 + \cos^{3} 3 x)}^{- \frac{4}{3}}

Step 3
Hence derivative of

y = {(3 + \cos^{3} 3 x)}^{- \frac{1}{3}} i s \frac{d y}{d x} = 3 \cos^{2} 3 x \sin 3 x {(3 + \cos^{3} 3 x)}^{- \frac{4}{3}}

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