Problem

Find the derivative of the following functions. $…

06:36

Problem 41 Hard Difficulty

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

Answer

$\dfrac{2sin x}{(1+cosx)^{2}}$

Related Courses

Calculus 1 / AB

Calculus Early Transcendentals

Chapter 3

Derivatives

Section 5

Derivatives of Trigonometric Functions

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Video Transcript

So this question asked us to find the derivative of y equals one minus co sign of X over one plus CO sign of X. And because this is a fraction, we're going to want to use a questionable which could be outlined in this red on the right. So to start off, we should identify our G and F terms. So our g of X is always gonna be within the denominator. So that's one plus co sign of X and then g Prime of X will therefore equal negative sign effects. Our affects, well, well, equal What's in the numerator. So one minus clear sign of acts and then at prime of X will equal find backs memory that we have this negative here. So the derivative of co sign is always negative. Sign that we have a negative co sign. It's going to equally positive sign. So now we can plug in our terms into the quotient rule. So we have one plus co sign of eggs times sign it. Eggs minus negative. Sign of X Times one minus co. Sign of X all over one plus co sign of X squared. So to start off, we're going to want to. You distribute this sign interim to both of our other terms, and I'm going to group these two terms in brackets just so we can move around these negative signs a little bit. But we're also going to want to distribute this sign to both terms. So when we do that now we get sign of eggs plus co sign of X Sign it. Eggs minus minus. Sign of eggs, plus cool sign of X, Sign it eggs oh, over one plus co sign of X squared. And now we want to distribute this negative sign people of our terms. So now we'll get sign of X plus co sign of X sign of Max Close sign of X minus cool sign of X sign of X all over one plus CO sign of X squared and then on the new reader. We can group together our like terms when we see the sign next term here in a sine x trail here in this used to terms happen to cancel out with each other. So in the end we are left with two sign of X over one plus co sign the X squared

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