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

Sum

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

#### 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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