# Find dydx, if y = xxx - Mathematics and Statistics

Sum

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)]

Concept: The Concept of Derivative - Derivatives of Logarithmic Functions
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Chapter 1.3: Differentiation - Q.5
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