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Question
Find `dy/dx` if y = xx + 5x
Solution
y = xx + 5x
let u= xx and v = 5x
∴ y = u + v
∴ `dy/dx = (du)/dx + (dv)/dx` (i)
Now u = xx
Taking logarithm on both the sides
log u = log xx
⇒ log u = x.log x
Differentiate w.r.t.x.
`1/u (du)/dx = x. 1/x + log x.1`
= 1 + log x
`therefore dy/dx = u( 1 + log x )`
`therefore du/dx = xx( 1 + log x )` (ii)
Now, v = 5x
`therefore dv/dx = 5x.log 5` (iii)
Substituting (ii) and (iii) in eqn (i)
`dy/dx = x^x ( 1 + log x ) + 5^x log 5`
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