a: \(y=3\cdot tanx-2\cdot cotx\)
=>\(y'=3\left(tan^2x+1\right)-2\cdot\left(-1\right)\cdot\left(cot^2x+1\right)\)
=>\(y'=3\cdot tan^2x+3+2\cdot cot^2x+2\)
=>\(y'=3\cdot tan^2x+2\cdot cot^2x+5\)
b: \(y=3\cdot cot\left(2x+\dfrac{\Omega}{3}\right)\)
=>\(y'=3\cdot\left(-1\right)\cdot\left(2x+\dfrac{\Omega}{3}\right)'\cdot\left(cot^2\left(2x+\dfrac{\Omega}{3}\right)+1\right)\)
=>\(y'=-6\cdot\left(cot^2\left(2x+\dfrac{\Omega}{3}\right)+1\right)\)
=>\(y'=-6\cdot cot^2\left(2x+\dfrac{\Omega}{3}\right)-6\)
c: \(y=tan\left(3x-4\right)+cot\left(x-5\right)\)
=>\(y'=\left(3x-4\right)'\cdot\left(tan^2\left(3x-4\right)+1\right)+\left(-1\right)\cdot\left(x-5\right)'\cdot\left(cot^2\left(x-5\right)+1\right)\)
=>\(y'=3\cdot\left(tan^2\left(3x-4\right)+1\right)-1\cdot\left(cot^2\left(x-5\right)+1\right)\)
=>\(y'=3\cdot tan^2\left(3x-4\right)-cot^2\left(x-5\right)+2\)
d: \(y=5\cdot tan\left(3x-\dfrac{\Omega}{5}\right)-3\cdot cot\left(2x+\dfrac{7}{6}\Omega\right)\)
=>\(y'=5\cdot\left(3x-\dfrac{\Omega}{5}\right)'\cdot\left(tan^2\left(3x-\dfrac{\Omega}{5}\right)+1\right)+3\cdot\left(2x+\dfrac{7}{6}\Omega\right)'\cdot\left(cot^2\left(2x+\dfrac{7}{6}\Omega\right)+1\right)\)
=>\(y'=15\left(tan^2\left(3x-\dfrac{\Omega}{5}\right)+1\right)+6\left(cot^2\left(2x+\dfrac{7}{6}\Omega\right)+1\right)\)
=>\(y'=15\cdot tan^2\left(3x-\dfrac{\Omega}{5}\right)+6\cdot cot^2\left(2x+\dfrac{7}{6}\Omega\right)+21\)
e: \(y=3\cdot tan^3x-cot^43x\)
=>\(y'=3\cdot3\cdot tan^2x\cdot\left(tanx\right)'-4\cdot cot^33x\cdot\left(cot3x\right)'\)
=>\(y'=9tan^2x\cdot\left(tan^2x+1\right)-4\cdot cot^33x\cdot\left(-1\right)\cdot\left(3x\right)'\cdot\left(cot^23x+1\right)\)
=>\(y'=9\cdot tan^4x+9tan^2x+12\cdot cot^33x\left(cot^23x+1\right)\)
=>\(y'=9\cdot tan^4x+9\cdot tan^2x+12\cdot cot^53x+12\cdot cot^33x\)
