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Question
Find `"dy"/"dx"` if y = xx + 5x
Solution
y = xx + 5x
Let u = xx , v = 5x
Taking log on both sides of u
log u = x log x
Dtffercnttallng w.r.t. x
`therefore 1/"u" "du"/"dx" = "x" (1/"x") + "log x" . (1)`
`therefore "du"/"dx" = "u" (1 + "log x")`
= xx (1 + log x)
Differentiating v. w.r.t. x
`"dv"/"dx"` = 5x log 5
As y = u + v
`therefore "dy"/"dx" = "du"/"dx" + "dv"/"dx"`
`therefore "dy"/"dx" = "x"^"x" (1 + "log x") + 5^"x" "log" 5`
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