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Find the derivative of the following functions.$$y=\frac{x \sin x}{1+\cos x}$$

Calculus 1 / AB

Chapter 3

Derivatives

Section 5

Derivatives of Trigonometric Functions

Differentiation

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Boston College

Lectures

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we are looking for the derivative of why is equal to ex side of X over one plus co side of X. The first rule we're going to use is the quotient rule, which is f over G prime is equal to F crime times G minus G prime times f All of this over G squared Right now we know that our f is equal Teoh x sign of X r g is equal to would plus co sign of X Now we're looking for our f prime and R g prime. So first up is finding our f prime To do that, it will look like D over DX of X side of X. To solve this, we're going to use our product rule f times g prime is equal Teoh f Prime times G plus f times Gede prime o r f is X so I f prime will be one RG is sign of X so our G prime will be co sign of X. We know that sign if X is equal to co sign of X through the common derivative identity already. So the next thing we're going to do is to combine all of the variables together to get one times side of X plus coastline of X times X simplified, it will look like side of X top plus x co side of X. This is our numerator. So now we're going to find the derivative of our denominator the over D X of wooden plus co sign of X to solve For this derivative, we're going to use the some difference rule F plus minus G prime is equal to F prime plus minus cheaper. Our rule applied will look like D over DX off one plus D over DX of co sign of X. So using our derivative d over DX of one is equal to zero. We know this through the derivative of constant. So any time you see a constant D over DX of a a being, any number you always get zero Next derivative is co sign of X D over DX of co Sign of X is equal to negative side of X. We know this through common drop common derivatives D over DX of coastline of X is always equal to negative Sign of x. Okay, so let's put all of this together we will get zero minus sign of X, which is equal to negative scientifics. All right, so now we know the derivatives of our numerator and denominator. Let's put it back into the variables we just learned earlier so our G prime would be negative. Sign of X and our f prime will be sign of X plus x Co sign of Lex. To put this all together we will get side of X plus x co sign if X multiplied by one plus co sign of X minus negative sign of X. Well, it's applied by x sign of sex. All of this is over. One plus co sign X squared. Simplified. We will get sign of X plus x Co Sign of X What's applied by one plus coastline of X close X sine squared of vets. All of this is over one plus co sign X squared. This is the final answer

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