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Question
If y = `log[4^(2x)((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)]`, find `("d"y)/("d"x)`
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Solution
y = `log[4^(2x)((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)]`
= `log4^(2x) + log((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)`
= `2x log4 + 3/2[log(x^2 + 5)/(sqrt(2x^3 - 4))]`
= `2x log4 + 3/2[log(x^2 + 5) - logsqrt(2^3 - 4)]`
= `2x log4 + 3/2[log(x^2 + 5) - 3/4log(2x^3 - 4)]`
Differentiating w. r. t. x, we get
`("d"y)/("d"x) = "d"/("dx)[2xlog 4 + 3/2 log(x^2 + 5) - 3/4log(2x^3 - 4)]`
= `2log4*"d"/("d"x)(x) + 3/2*"d"/("d"x) [log(x^2 + 5)] - 3/4*"d"/("d"x) [log(2x^3 - 4)]`
= `2log4*1 + 3/*1/(x^2 + 5)*"d"/("d"x) (x^2 + 5) - 3/4*1/(2x^3 - 4)*"d"/("d"x)(2x^3 - 4)`
= `2log4 + 3/2*1/(x^2 + 5)*2x - 3/4*1/(2x^3 - 4)*6x^2`
∴ `("d"y)/("d"x) = 2log4 + (3x)/(x^2 + 5) - (9x^2)/(2(2x^3 - 4)`
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