That's really a complicated one......
I will give it a go.
\(f(x)=\dfrac{3^x\ln x\sqrt{(\ln x)(x^{\sqrt{3}-1})}}{2^x\sin x \cot x}=(\dfrac{3}{2})^x(\dfrac{\ln x\sqrt{(\ln x)(x^{\sqrt3-1})}}{\cos x})\)
\(=(\dfrac{3}{2})^x(\dfrac{\ln x}{\cos x})(\sqrt{(\ln x)(x^{\sqrt3-1})})\)
There we seperate it into 3 parts!!
Part 1 = (3/2)^x
Part 2 = (ln x)/(cos x)
Part 3 = sqrt((ln x)(x^(sqrt 3 - 1)))
Next we will find the derivative of (3/2)^x using implicit differentiation......
\(y=(\dfrac{3}{2})^x\\ \ln y = x\ln \dfrac{3}{2}\\ \dfrac{1}{y}\dfrac{dy}{dx}=\ln\dfrac{3}{2}\\ \dfrac{dy}{dx}=y\ln\dfrac{3}{2}=(\dfrac{3}{2})^x(\ln\dfrac{3}{2})\)
Then we find the derivative of ln x/ cos x......
\((\dfrac{\ln x}{\cos x})'\\ = \dfrac{(\frac{1}{x})(\cos x)-(-\sin x)(\ln x)}{\cos^2 x}\\ =\dfrac{\frac{\cos x}{x}+\sin x \ln x}{\cos^2x}\\ =\dfrac{1}{x\cos x}+\dfrac{\tan x \ln x}{\cos x}\\ =\dfrac{1+x\tan x \ln x}{x\cos x}\)
Then we came to the most complicated part: derivative of sqrt((ln x)(x^(sqrt 3 - 1)))
\((\sqrt{(\ln x)(x^{\sqrt3-1})})'\\ y=\sqrt{u}\\ u = (\ln x)(x^{\sqrt3-1})\\ y' = \dfrac{({x^{\sqrt3-2}})+((\sqrt3-1)(\ln x)(x^{\sqrt3-2}))}{2\sqrt{\ln x(x^{\sqrt3 -1})}}\)
Next we apply the product rule to Part 1 and Part 2 and call it Part 1,2
(Part 1 times Part 2)'
= Part 1 ' Part 2 + Part 2 ' Part 1
= \(\left(\left(\dfrac{3}{2}\right)^x\ln \dfrac{3}{2}\right)\left(\dfrac{\ln x}{\cos x}\right)+\left(\dfrac{1+x\tan x \ln x}{x\cos x}\right)\left(\dfrac{3}{2}\right)^x\\ \)
= \(\left(\dfrac{3^x\ln x\ln 3 -3^x\ln x \ln 2}{2^x\cos x}\right)+\left(\dfrac{3^x+3^x\cdot x\tan x \ln x}{2^x x\cos x}\right)\)
=\(\dfrac{3^x(x\ln x \ln 3 - x \ln x \ln 2+1+x\tan x \ln x)}{2^x x \cos x}\)
I will name this fraction y.
Then we find the derivative of Part1,2:
\(\dfrac{3^x \ln x(x\ln x \ln 3 - x \ln x \ln 2 + 1 + x \tan x \ln x)+3^x(\ln x \ln 3 + \ln 3 -\ln x \ln 2 - \ln 2+\tan x \ln x+x\sec^2x\ln x+\tan x)}{2^{2x}x^2\cos^2x}\)Oops it's too long :( I will name this looooooonnnnnggggg fraction z then)
Then we use product rule for Part1,2 and Part 3:
f'(x)= \(y(\dfrac{x^{\sqrt3-2}+(\sqrt3-1)(\ln x)(x^{\sqrt3-2})}{2\sqrt{\ln x(x^{\sqrt3-1})}})+z(\sqrt{\ln x(x^{\sqrt3-1})})\)
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