a: \(y'=\dfrac{\left(x^2+3x-1\right)'\cdot\left(x+2\right)-\left(x^2+3x-1\right)\cdot\left(x+2\right)'}{\left(x+2\right)^2}\)
\(=\dfrac{\left(2x+3\right)\left(x+2\right)-\left(x^2+3x-1\right)}{\left(x+2\right)^2}\)
\(=\dfrac{2x^2+7x+6-x^2-3x+1}{\left(x+2\right)^2}=\dfrac{x^2+4x+7}{\left(x+2\right)^2}\)
b: \(y'=\dfrac{\left(2x^2-x\right)'\cdot\left(x^2+1\right)-\left(2x^2-x\right)\left(x^2+1\right)'}{\left(x^2+1\right)^2}\)
\(=\dfrac{4x\left(x^2+1\right)-2x\left(2x^2-x\right)}{\left(x^2+1\right)^2}\)
\(=\dfrac{4x^3+4x-4x^3+2x^2}{\left(x^2+1\right)^2}=\dfrac{2x^2+4x}{\left(x^2+1\right)^2}\)
c: \(\left(\dfrac{3-2x}{x-1}\right)'=\dfrac{\left(3-2x\right)'\left(x-1\right)-\left(3-2x\right)\left(x-1\right)'}{\left(x-1\right)^2}\)
\(=\dfrac{-2\left(x-1\right)-\left(3-2x\right)}{\left(x-1\right)^2}=\dfrac{-2x+2-3+2x}{\left(x-1\right)^2}=-\dfrac{1}{\left(x-1\right)^2}\)
\(\left(\sqrt{2x-3}\right)'=\dfrac{\left(2x-3\right)'}{2\sqrt{2x-3}}=\dfrac{1}{\sqrt{2x-3}}\)
\(y'=\left(\dfrac{3-2x}{x-1}\right)'+\left(\sqrt{2x-3}\right)'\)
\(=\dfrac{-1}{\left(x-1\right)^2}+\dfrac{1}{\sqrt{2x-3}}\)
