Find dydxif, y = (x)x+(ax) - Mathematics and Statistics

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Sum

Find `"dy"/"dx"`if, y = `("x")^"x" + ("a"^"x")`

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Solution

y = `("x")^"x" + ("a"^"x")`

Let u = `("x")^"x"` and v = `("a"^"x")`

∴ y = u + v

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

`"dy"/"dx" = "du"/"dx" + "dv"/"dx"`     ....(i)

Now u = `("x")^"x"`

Taking logarithm of both sides, we get

log u = log `("x")^"x"`

∴ log u = `"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"`

∴ `"du"/"dx" = "u"(1 + log "x")`

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

v = ax

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

`"dv"/"dx" = "a"^"x"* log "a"`       ....(iii)

Substituting (ii) and (iii) in (i), we get

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

Concept: The Concept of Derivative - Derivatives of Logarithmic Functions
  Is there an error in this question or solution?
Chapter 3: Differentiation - Exercise 3.3 [Page 94]

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