What is the derivative of \frac{2xsinx}{2+cox}

Question:

What is the derivative of {eq}\frac{2x sin x}{2 + cos x} {/eq}

Differentiation Strategy:

The derivative of the trigonometric terms and the algebraic terms in composite expression form can be found using the quotient rule and the product rule of differentiation. The quotient rule is given as: {eq}\left(\frac{f}{g}\right)'=\frac{f\:'\cdot g-g'\cdot f}{g^2} {/eq}

Answer and Explanation: 1


The derivative of the expression is given as:

{eq}y'=\frac{d}{dx}\left(\:\frac{2x\:sin\:x}{2\:+\:cos\:x}\right)\\ \mathrm{Apply\:the\:Quotient\:Rule}:\quad \left(\frac{f}{g}\right)'=\frac{f\:'\cdot g-g'\cdot f}{g^2}\\ =2\cdot \frac{\frac{d}{dx}\left(x\sin \left(x\right)\right)\left(2+\cos \left(x\right)\right)-\frac{d}{dx}\left(2+\cos \left(x\right)\right)x\sin \left(x\right)}{\left(2+\cos \left(x\right)\right)^2}\\ {/eq}

Now the derivative of:

{eq}\frac{d}{dx}\left(x\sin \left(x\right)\right)=\frac{d}{dx}\left(x\right)\sin \left(x\right)+\frac{d}{dx}\left(\sin \left(x\right)\right)x\\ =\sin \left(x\right)+x\cos \left(x\right)\\ {/eq}

and the derivative of :

{eq}\frac{d}{dx}\left(2+\cos \left(x\right)\right)=-\sin \left(x\right)\\ {/eq}

Thus we get the final expression for the derivative, as follows:

{eq}y'=2\cdot \frac{\left(\sin \left(x\right)+x\cos \left(x\right)\right)\left(2+\cos \left(x\right)\right)-\left(-\sin \left(x\right)\right)x\sin \left(x\right)}{\left(2+\cos \left(x\right)\right)^2}\\ =\frac{2\left(\left(\sin \left(x\right)+x\cos \left(x\right)\right)\left(2+\cos \left(x\right)\right)+x\sin ^2\left(x\right)\right)}{\left(2+\cos \left(x\right)\right)^2} {/eq}


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Differentiation Strategy: Definition & Examples

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Chapter 7 / Lesson 15
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In this lesson, we'll learn about differentiation strategy. We'll define it and look at important characteristics. The lesson will then discuss the pros and cons of differentiation strategy.


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