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Question
Find `("d"y)/("d"x)`, if y = `x^(x^x)`
Solution
y = `x^(x^x)`
Taking logarithm of both sides, we get
log y = `log x^(x^x)`
∴ log y = xx log x
Differentiating both sides w.r.t. x, we get
`"d"/("d"x)(log y) = x^x*"d"/("d"x)(log x) + logx*"d"/("d"x)(x^x)`
∴ `1/y*("d"y)/("d"x) = x^x*1/x + logx*"d"/("d"x)(x^x)` ......(i)
Let u = xx
Taking logarithm of both sides, we get
log u = log xx
∴ log u = x log x
Differentiating both sides w.r.t. x, we get
`"d"/("d"x)(log "u") = x*"d"/("d"x)(log x) + logx*"d"/("d"x)(x)`
∴ `1/"u"*"du"/("d"x) = x*1/x + logx*1`
∴ `1/"u"*"du"/("d"x)` = 1 + log x
∴ `"du"/("d"x)` = u(1 + log x)
∴ `"d"/("d"x)(x^x)` = xx(1 + log x) ......(ii)
Substituting (ii) in (i), we get
`1/y*("d"y)/("d"x) = x^x*1/x + logx*x^x(1 + log x)`
∴ `("d"y)/("d"x) = yx^x[1/x + logx(1 + logx)]`
∴ `("d"y)/("d"x) = x^(x^x)*x^x[1/x + logx(1 + logx)]`
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