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Question
If y = 5x + xx, Find `(dy)/(dx)`.
Solution
y = 5x + xx
Let u = 5x and v = xx
y = u + v
Differentiating both sides w.r.t.x
`(dy)/(dx) = (du)/(dx) + (dv)/(dx)`
`(du)/(dx)` = 5x. log5
Now. v = xx
Taking log on both sides
log v = log(xx)
log v= x. log x
`1/v (dv)/(dx) = x. 1/x` + log x. 1
`therefore (dv)/(dx)` = v [1 + log x]
`therefore (dv)/(dx) = x^x [1 + log x]`
`(dy)/(dx) = (du)/(dx) + (dv)/(dx)`
`therefore (dy)/(dx)` = 5x . log 5 + xx [1 + logx]
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