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महाराष्ट्र राज्य शिक्षण मंडळएचएससी विज्ञान (सामान्य) इयत्ता १२ वी

If y = 5x. x5. xx. 55 , find dydx

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प्रश्न

If y = 5x. x5. xx. 55 , find `("d"y)/("d"x)`

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उत्तर

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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