फलन ` (sin x)^(cos x) + ( cos x)^(sin x)` का x के साक्षेप अवकल गुणांक ज्ञात कीजिए ।

Explanation

मान लो ` u = ( sin x ) ^(cos x) `
` :. log_ (e) u = cos . log_(e) sin x `
` :. 1/u (du)/(dx) = cos x. 1/sin x . cos x + ( log _(e) sin x ) ( -sin x ) `
या ` (du)/(dx) = u [ (cos^(2)x)/(sin x) - sin x log _(e) sin x ] `
` = ( sin x) ^(cos x ) [ ( cos^(2) x )/(sin x) - sin x log _(e) sin x ]`
पुनः मानलो ` v = ( cos x )^(sin x) `
` :. log v = sin x log _(e) cos x `
` 1/v (dv)/(dx) = sin x 1/ cos x (-sin x) + ( log_(e) cos x ) . cos x `
या ` (dv)/(dx) = v [ -(sin^(2)x)/(cos x) + cos x log_(e) cos x ] `
` ( cos x ) ^(sin x ) ( cos x log _(e) cos x - (sin^(2) x)/(cos x ) )`
` :. अभीष्ट अवकल गुणांक = (du)/(dx) + ( dv) / (dx) `
` ( sin x) ^(cos x)(( cos^(2) x)/(sin x) - sin x log _(e) sin x ) + (cos x) ^(sin x) ( cosx log_(e) cos x - ( sin^(2) x)/(cos x))`.