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

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

Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 2

Derivatives of the Other Trigonometric Functions

Derivatives

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we want to find the derivative dy dx for the function Why is equal to tangent squared of three X over two plus seven X squared plus one. To solve this problem, we're going to rely on the derivative shortcuts. We picked up through a single variable crack in particular. We're going to rely on the trigonometry derivatives that we've learned recently. So these are the trick derivatives. DDX Synnex coast equals Cossacks BD, exposed to the negative syntax and so on. We particularly want to use a tangent derivative and sign derivative given this function. I've also ordered the chain rule and product rule, both of which will come in handy for this problem. So let's rewrite why as tan square three X times two plus sine X squared plus one is negative first. This means we can start off the derivative of the product rule. So you I D. X equals to tan three X. C. Can't square three X times three times two plus seven X. Where the negative. First this is our derivative plans were three X plus 10 square three X times negative. Two plus two plus one is negative second times cosine X squared plus one times two X. This applies the chain rule on the two plus seven X square plus one of the first term. Plus. We have the final solution given in yellow at the bottom.

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