Find `(dy)/(dx),` `y=(sinx)^(cosx)+(cosx)^(sinx)`

Explanation

`y=(sin x)^{cos x}+(cos x)^{sin x}`

Taking log both side and we get,

`log y=log (sin x)^{cos x}+log (cos x)^{sin x}`

`log y=cos x cdot log sin x+sin x cdot log cos x`

`frac{d}{d x} log y=cos x frac{d}{d x} log sin x+log sin x frac{d}{d x} cos x+sin x frac{d}{d x} log cos x+`
`log cos x frac{d}{d x} sin x`

`Rightarrow frac{1}{y} frac{d y}{d x}=cos x frac{1}{sin x} cos x+log sin x(-sin x)+sin x frac{1}{cos x}(-sin x)+log cos x cos x`

`Rightarrow frac{1}{y} frac{d y}{d x}=frac{cos ^{2} x}{sin x}-sin x log sin x-frac{sin ^{2} x}{cos x}+cos x log cos x `

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