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Question
If y = 5x. x5. xx. 55 , find `("d"y)/("d"x)`
Solution
y = 5x. x5. xx. 55
Taking log on both sides, we get
log y = log(5x. x5. xx. 55)
= log 5x + log x5 + log xx + log 55
∴ log y = xlog 5 + 5 log x + xlog x + 5log 5
Differentiating w.r.t. x, we get
`"d"/("d"x)(log y) = "d"/("d"x)(x log 5 + 5 log x + x log x + 5 log 5)`
∴ `1/y*("d"y)/("d"x) = log5*"d"/("d"x)(x) + 5*"d"/("d"x)(log x) + x*"d"/("d"x)(log x) + logx* "d"/("d"x)(x) + "d"/("d"x)(5log5)`
= `log5*1 + 5*1/x + x*1/x + logx*1 + 0`
∴ `("d"y)/("d"x) = y(log5 + 5/x + 1 + logx)`
∴ `("d"y)/("d"x) = 5^x* x^5* x^x* 5^5 (log5 + 5/x + 1 + logx)`
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