1.

यदि ` y= (sin x )^(x) +sin ^(-1) sqrt (x) ,` तब ` (dy)/(dx) ` का मान ज्ञात कीजिए|

Answer» यहाँ ` " "y= (sin x ) ^(x) +sin^(-1) sqrt(x) `
माना, ` " "y_1 =(sinx )^(x) ` व ` y_2 =sin ^(-1) sqrt x` तब
` " " y=y_1 +y_2 `
` rArr " " (dy)/(dx) =(dy_1)/(dx) +(dy_2)/(dx) " "....(1)`
` " "y_1 =(sin x ) ^(x) `
` " "logy_1 =x log (sinx )`
x के सापेक्ष अवकलन करने पर
` (1)/(y_1) *(dy_1)/(dx) =x* (1)/(sin x ) *cos x+ log (sim x )*1 `
` " "= (sin x )^(x) [x cot x +log (sin x)]" "....(2) `
अब, ` " "y_2 = sin ^(-1) sqrt x `
x के सापेक्ष अवकलन करने पर
` " "(dy_2)/(dx) =(1)/(sqrt( 1-x) )*(1)/(2) x^(1//2) =(1)/(sqrt(x-x^(2))) `
समीकरण (1 ),(2 ) व (3 ) से,` (dy)/(dx)=(sin x )^(x) {x cot x + log (sin x )} +(1)/(2sqrt (x-x^(2)) )`


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