a: \(y=\frac{1+\sin x}{2-\sin x}\)
=>y'=\(\frac{\left(1+\sin x\right)^{\prime}\left(2-\sin x\right)-\left(1+\sin x\right)\left(2-\sin x\right)^{\prime}}{\left(2-\sin x\right)^2}\)
\(=\frac{cosx\left(2-\sin x\right)+cosx\left(1+\sin x\right)}{\left(2-\sin x\right)^2}=\frac{3\cdot cosx}{\left(2-\sin x\right)^2}\)
b: \(y=\frac{\sin x+cosx}{\sin x-cosx}\)
=>y'\(=\frac{\left(\sin x+cosx\right)^{\prime}\left(\sin x-cosx\right)-\left(\sin x+cosx\right)\left(\sin x-cosx\right)^{\prime}}{\left(\sin x-cosx\right)^2}\)
\(=\frac{\left(cosx-\sin x\right)\left(\sin x-cosx\right)-\left(cosx+\sin x\right)\left(cosx+\sin x\right)}{\left(\sin x-cosx\right)^2}\)
\(=\frac{-\left(\sin x-cosx\right)^2-\left(\sin x+cosx\right)^2}{\left(\sin x-cosx\right)^2}=\frac{-\left(1-\sin2x\right)-\left(1+\sin2x\right)}{1-\sin2x}=\frac{-1+\sin2x-1-\sin2x}{1-\sin2x}=\frac{-2}{1-\sin2x}\)
c: y=5sin x-3*cosx
=>y'=\(5\cdot\left(\sin x\right)^{\prime}-3\cdot\left(cosx\right)^{\prime}\)
=>y'=5cosx+3sinx
