Calculating derivatives Find dy&gt;dx for the following functions.y=1−cos⁡x1+cos⁡x

Step 1Use the identities cos⁡0=1−2sin20=2cos20−1 and simplify the rational function.Step 2y=1−(1−2sin2x)1+(2cos2x−1)=2sin2x2cos2x=tan2x (tan⁡x=sin⁡xcos⁡x)Step 3Now differentiate the function y=tan2x with respect to x.dydx=2tan⁡xsec2xStep 4Conclusion:dydx=2tan⁡xsec2x

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Answered question

2022-03-24

Calculating derivatives Find dy>dx for the following functions.

y = \frac{1 - \cos x}{1 + \cos x}

Answer & Explanation

Sawyer Anthony

Beginner2022-03-25Added 10 answers

Step 1
Use the identities

\cos 0 = 1 - 2 \sin^{2} 0 = 2 \cos^{2} 0 - 1

and simplify the rational function.
Step 2

y = \frac{1 - (1 - 2 \sin^{2} x)}{1 + (2 \cos^{2} x - 1)}

= \frac{2 \sin^{2} x}{2 \cos^{2} x}

= \tan^{2} x (\tan x = \frac{\sin x}{\cos x})

Step 3
Now differentiate the function

y = \tan^{2} x

with respect to x.

\frac{d y}{d x} = 2 \tan x \sec^{2} x

Step 4
Conclusion:

\frac{d y}{d x} = 2 \tan x \sec^{2} x

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