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Find the derivatives of the given functions.$$y=\frac{\cos ^{2} 3 x}{1+2 \sin ^{2} 2 x}$$

$\frac{d y}{d x}=\frac{6 \cos 3 x \sin 3 x\left(1+2 \sin ^{2} 2 x\right)-8 \cos ^{2} 3 x \sin 2 x \cos 2 x}{\left(1+2 \sin ^{2} 2 x\right)^{2}}$

Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 1

Derivatives of the sine and Cosine Functions

Derivatives

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All right. We want to find Dy DX for the function. Why is equal to the coast Sine squared of three X divided by one plus two Sine squared two X. To do so, we're going to rely on the road and shortcuts we picked up throughout single variable help particularly rule 033 With that here Roll zero is DDX in Mexico's coast sex. Tv expose exploitative science rules one through three other power world product rule in general, respectively. If we write why in a more easily differential form, we'll see how we can differentiate. Does he write why? As co squared three X times one plus two Sine squared two X 91st. Now we see how the derivative. It started off with the products that we proceeded. The derivative Dy DX is six coastline three extends to sign three X over one plus two Sine squared two X. This applies rules zero and three to co square three X minus coast squared three X times four signed two X times to co sign two X over one plus two Sine squared two X squared. This time we apply rule 0 to 3 again to one plus to sign to square two X two naked first, multiplying the first term. Buy one plus to sign two X over one plus to sign 22 X gives final solution. Dy dx is equal to six coastline three X sign three X one plus two Sine squared two x minus eight. Post square three X sign too exposed to X. All over one plus two Sine squared two X over squared

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