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Find the derivatives of the given functions.$$y=3 \sin ^{3}\left(2 x^{4}+1\right)$$

$\frac{d y}{d x}=-15 \cos ^{2}(5 x+2) \sin (5 x+2)$

Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 1

Derivatives of the sine and Cosine Functions

Derivatives

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uh huh. We want to find dy dx for Y is equal to three kind of sine cubed, two extra, fourth plus one to do so we rely on derivative shortcuts we picked up throughout single calculus that is single variable calculus excuse me. Combined with specifically trigger metric derivatives. So first we have relevant rules listed here on the last zero DDX. In Mexico Cossacks. DDX Cossacks is negative syntax. We also have rules one through three listed where one is the power rule to the product rule through the chain rule. In this particular problem we're gonna have to use rule +01 and three to solve. So first we write our derivative in a way that's easy to solve for. So the function Y equals three signed two X to the fourth plus one cubed. This makes it easy to use the train rule. Thus we have dy dx is three times to sign square. Two extra. The fourth plus one times the derivative was inside which is co signed to actually the fourth plus one times eight X cube. Giving a final solution. DY Dx equals 48 X cube. Sine squared two extra. The fourth plus one, co signed to actually the fourth plus one.

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