Practice Find derivative of the sum of two quotients

anemone

Paris la ville de l'amour
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#1
Find the derivative of \(\displaystyle \dfrac{\sin^2 x}{1+\cot x}+\dfrac{\cos^2 x}{1+\tan x}\).
 
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MarkFL

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#2
I think I would begin by looking at ways to simplify first, before differentiating:

\(\displaystyle f(x)=\frac{\sin^2(x)}{1+\cot(x)}+\frac{\cos^2(x)}{1+\tan(x)}=\frac{\sin^3(x)}{\sin(x)+\cos(x)}+\frac{\cos^3(x)}{\cos(x)+\sin(x)}=\frac{\sin^3(x)+\cos^3(x)}{\sin(x)+\cos(x)}=\sin^2(x)+\sin(x)\cos(x)+\cos^2(x)=1+\frac{1}{2}\sin(2x)\)

And so:

\(\displaystyle f'(x)=\cos(2x)\)
 
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anemone

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#3
Very well done, Mark!

This problem is hard to differentiate if one opts to start differentiating and then simplified later...but if we simplify it first, then what we need to differentiate is only the sine function.

Thanks for participating, Mark!
 
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