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

Explanation

माना ` " " y= (sin x ) ^(cos x) +(cos x)^(sin x) `
punha माना ` " "u= (sin x )^(*cos x ) `
तथा ` " "v= (cos x )^(sin x) `
` rArr " "(dy)/(dx) =(du)/(dx) + (dv)/(dx) " "...(1)`
अब ` " "u =(sin x ) ^(cos x) `
` rArr " "logu =cos x log (sin x ) `
दोनों पक्षों का x के सापेक्ष अवकलन करने पर
` rArr (1)/(u) (du)/(dx) =cos x (d)/(dx) (log sin x )+ log sin x(d)/(dx) (cos x ) `
` rArr" "(du)/(dx) =u (cos x ""(1)/(sin x ) *cos x +log sin x *(-sin x )) `
` " "= (sin x )^(cosx) (cos x cot x -sin x log sin x )" "...(2)`
isi prakar ` " "v =(cos x )^(sinx ) `
` rArr " " log v =sin x log cos x `
दोनों पक्षों का x के सापेक्ष अवकलन करने पर
` rArr " "(1)/(v) (dv)/(dx) =sin x (1)/(cos x)(-sin x )+ (log cos x ) cos x `
` rArr " "(dv)/(dx) =v [-sin x tan x + (log cos x )cos x ]`
` " "=(cos x)^(sin x ) [cos x log cos x -sin x tan x ]`
समीकरण (1 ) ,(2 ) व् (3 ) से,
` (dy)/(dx) =(sin x)^(cos x ) (cos x cot x - sin x log sin x ) `
` " "+ (cos x )^(sin x ) (cos xlog cos x -sin x tan x )`