फलन `(sin x)^(cos x)+(cos x)^(sin x)` का x के सापेक्ष अवलन गुणांक ज्ञात कीजिये ।

Explanation

`y= (sin x)^(cos x) + ( cos x ) ^(sin x)`
या ` y = u + v `
जहाँ `u=(sin x ) ^(cos x)`
और `v=(cos x)^(sin x)`
` :. (dy)/(dx) = (du)/(dx) + (dv)/(dx)" "`…(i)
अब ` u = (sin x) ^(cos x)`
या `log u = cos x log sin x `
या ` 1/u (du)/(dx)= cos x *(1/sinx* cos x)`
` + (log sin x ) (- sin x )`
` = (cos^(2) x )/(sin x )- sin x log sin x `
या ` (du)/(dx) = u [(cos^(2)x)/(sin x) - sin x log sin x ]`
या `(du)/(dx)= (sin x ) ^(cos x)`
`[ (cos^(2) x)/(sin x ) - sin x log sin x ]" "` ...(ii)
और ` v = (cos x ) ^(sin x)`
या ` log v = sin x log cos x `
या ` 1/v (dv)/(dx) = sin x * [ 1/(cos x ) * (- sin x )]`
` + (log cos x ) (cos x )`
` = - (sin^(2) x )/(cos x ) + cos x log cos x `
या ` (dv)/(dx) = v [ -(sin^(2))/(cos x )+ cos x log cos x ]`
या ` (dv)/(dx) + (cos x) ^(sin x )`
` [ - (sin^(2)x)/(cos x ) + cos x log cos x ]" " ` ...(iii)
समीकरण (i) व (ii) से ,
` (dy)/(dx) = ( sin x ) ^(cos x) [ (cos^(2) x ) /(sin x) - sin x log sin x ]`
` + (cos x ) ^(sin x ) [-(sin^(2) x)/(cos x ) + cos x log x ]`