How do you differentiate #f(x)=x / cot (x) + 3# using the quotient rule?

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Answer 1 · 2016-05-28T09:08:11

#\frac{x+\sin ^2t(x)\cot (x)}{\sin ^2\(x)\cot ^2\(x)}#

#\frac{d}{dx}(\frac{x}{\cot (x)}+3\)#

Applying sum/difference rule,#\(f\pm g\)^'=f^'\pm g^'#

#=\frac{d}{dx}\(\frac{x}{\cot t(x)})+\frac{d}{dx}(3)#

Now,
#=\frac{d}{dx}\(\frac{x}{\cot t(x)})#
Applying quotient rule,#(\frac{f}{g})^'=\frac{f^'\cdot g-g^'\cdot f}{g^2}#

#=\frac{\frac{d}{dx}(x)\cot (x)-\frac{d}{dx}(\cot (x))x}{\cot ^2(x)}#

And,we know,#\frac{d}{dx}(x)=1# and, #\frac{d}{dx}(\cot (x))=-\frac{1}{\sin ^2(x)}#

Also,#\frac{d}{dx}(3)=0#

Finally,
#=\frac{\sin ^2(x)\cot(x)+x}{\sin ^2\(x)\cot ^2(x)}+0#

Simplifying it,
#\frac{x+\sin ^2t(x)\cot (x)}{\sin ^2\(x)\cot ^2\(x)}#