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Differentiate sin(x^x) with respect to x...

Differentiate `sin(x^x)` with respect to `x` :

Text Solution

Verified by Experts

Given that,
`y=sin(x^x)`
`(dy)/(dx)=d/(dx)sin(x^x)`
`=>(dy)/(dx)=cos(x^x)*d/(dx)(x^x)`
let `t=x^x`
`=>log t= log(x^x)`
`=>log t= x log x`
`=>d/(dx) logt=d/(dx)(x log x)`
`=>1/t (dt)/(dx)=x*d/(dx)(log x)+log x*d/(dx) (x)`
`=>(dt)/(dx)=t[x*1/x+logx]`
`=>(dt)/(dx)=x^x[logx+1]`
now,`d/(dx)sin(x^x)=cos(x^x)*x^x(logx+1)`
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