# Find dy/dx if y = e^x^x - Mathematics and Statistics

Sum

Find "dy"/"dx"if, y = "e"^("x"^"x")

#### Solution

y = "e"^("x"^"x")

Taking the logarithm of both sides, we get

log y = log "e"^("x"^"x") = "x"^"x" log "e"

∴ log y = "x"^"x"

Differentiating both sides w.r.t.x, we get

1/"y" * "dy"/"dx" = "d"/"dx" ("x"^"x")    .....(i)

Let u = "x"^"x"

Taking logarithm of both sides, we get

log u = log "x"^"x" = "x" log "x"

Differentiating both sides w. r. t. x, we get

1/"u" * "du"/"dx" = "x" * "d"/"dx" (log "x") + log "x" * "d"/"dx"("x")

∴ 1/"u" * "du"/"dx" = "x" * 1/"x" + log "x" * (1)

∴ 1/"u" * "du"/"dx" = 1 + log x

∴ "du"/"dx" = "u"(1 + log "x")

∴ "du"/"dx" = "x"^"x"(1 + log "x")    .....(ii)

Substituting (ii) in (i), we get

1/"y" * "dy"/"dx" = "x"^"x"(1 + log x)

∴ "dy"/"dx" = "y"  "x"^"x" (1 + log "x") = "e"^("x"^"x") * "x"^"x" (1 + log "x")

Concept: The Concept of Derivative - Derivatives of Logarithmic Functions
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Chapter 3: Differentiation - Exercise 3.3 [Page 94]

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