Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12
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Find dydx, if y = 2xx. - Mathematics

Sum

Find `"dy"/"dx"`, if y = `2^("x"^"x")`.

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Solution

y = `2^("x"^"x")`

Taking logarithm of both sides, we get

`"dy"/"dx" = 2^("x"^"x") * log 2 * "d"/"dx" ("x"^"x")`          ....(i)

Let u = xx

log u = log (xx)

∴ log y = 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")`

`= "x" * 1/"x" + log "x" (1)`

∴ `1/"u" * "du"/"dx" = 1 + log "x"`

∴ `"dy"/"dx" = "u"(1 + log "x")`

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

Substituting (ii) in (i), we get

`"dy"/"dx" = 2^("x"^"x") * log 2 * "x"^"x" (1 + log "x")`

`"dy"/"dx" = 2^("x"^"x") * "x"^"x" * log 2 (1 + log "x")`

  Is there an error in this question or solution?
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APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 3 Differentiation
Miscellaneous Exercise 3 | Q 4.07 | Page 100
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