How do you differentiate given #y = 2x (x^(1/2) - cot x)#?

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Answer 1 · 2016-06-03T09:58:25

#=frac{2x+3sqrt{x}sin ^2(x)-2sin ^2(x)cot (x)}{sin ^2(x)}#

#frac{d}{dx}(2x(sqrt{x}-cot (x)))#

Taking the constant out,
#(a\cdot f)^'=a\cdot f^'#

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

Applying the product rule,
#(fcdot g)^'=f^'cdot g+fcdot g^'#
#f=x:g=sqrtx -cotx#

#=2(frac{d}{dx}(x)(sqrt{x}-cot(x))+frac{d}{dx}(sqrt{x}-cot (x))x)#

We know,
#frac{d}{dx}(x)=1#

#frac{d}{dx}(sqrt{x}-cot(x))=frac{1}{2sqrt{x}}+frac{1}{sin ^2(x)}#

#=2(1(sqrt{x}-\cot (x))+(frac{1}{2sqrt{x}}+frac{1}{sin ^2(x)})x)#

Simplify,
#=frac{2x+3sqrt{x}sin ^2(x)-2sin ^2(x)cot (x)}{sin ^2(x)}#