1.

यदि (If) `y=(sinx)^(x)+(cosx)^(tanx)`, (find) `(dy)/(dx)` निकालें

Answer» `y=(sinx)^(x)+(cosx)^(tanx)`
`=u+v` जहाँ `u=(sinx)^(x) ,v=(cosx)^(tanx)` … (1)
अब `u=(sinx)^(x)`
`rArrlogu=xlogsinx`
`rArr(1)/(u)(du)/(dx)=1*logsinx+x*cotx`
`rArr(du)/(dx)=(sinx)^(x)[logsinx+xcotx]`
पुनः `v=(cosx)^(tanx)`
`rArrlogv=tanxlogcosx`
`rArr(1)/(v)(dv)/(dx)=sec^(2)xlogcosx+tanx*(1)/(cosx)(-sinx)`
`=sec^(2)xlogcosx-tan^(2)x`
`rArr(dv)/(dx)=(cosx)^(tanx)[sec^(2)xlogcosx-tan^(2)x]` ... (2)
अब y=u+v
`therefore(dy)/(dx)=(du)/(dx)+(du)/(dx)`
`=(sinx)^(x)(logsinx+xcotx)+(cosx)^(tanx)[sec^(2)xlogcosx-tan^(2)x]`


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