Problem

Find the derivative of the following functions. $…

06:05

Problem 29 Medium Difficulty

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

Answer

$-\dfrac{1}{1+sinx}$

Related Courses

Calculus 1 / AB

Calculus Early Transcendentals

Chapter 3

Derivatives

Section 5

Derivatives of Trigonometric Functions

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Video Transcript

where this question we're asked to find the derivative of co sign over X over sign of X plus one. And because this is a fraction, we have to use the quotient rule, which can be summarized on the right hand side, inbred. So to begin, we're going to define our A and B terms. So our a well equal co sign of X and a prime will therefore equal negatives. Sign of X and our B who equals Sign of X Plus One and Be prime will therefore equal co sign of X. Remember that the derivative of one is zero, so we don't need to write that. So to begin, we're going to plug everything into the question rule, so we'll have co sign of Ex Prime Times. Sign of X plus one minus co sign of X Times, the derivative of sign of X plus one All over Sign of X plus one squared. So now we and simplify our terms so we'll get negative. Sign of X Times sign of X plus one minus co sign of X times. Look who sign of X all over Sign of X plus one squared. And from here we can distribute this negative sign of X to these teeth thes two terms and then distribute that coastline of X to the other co sign of X. So this will now simplify too. Negative Sign of X minus Sign squared of X minus Go sign of X again Over Nope. Sorry. This is co sine squared of ex all over sine X plus one squared And now I'm going to group these to sign terms together and distribute that negative sign. So now we'll have negative sign of X plus sine squared of X minus co sine squared of X all over, Sign in X plus one squared again I'm going to group my sign terms together and you can distribute that negative sign to you The co sine squared So now we have sign of X plus sine squared of x plus co sine squared of X all over, Sign of X plus one squared in this right here is a Trigon Amit Identity. The science world of explosive coast sine squared of X equals one. So now this simplifies to sign of X plus one over sign of X plus one squared. And because we have the same term in the numerator and denominator, we can cancel out the numerator with one of the exponents and the denominator to simplify our solution to one over sign of X plus one.

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