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Question
Differentiate the following w.r.t.x: `log[4^(2x)((x^2 + 5)/(sqrt(2x^3 - 4)))^(3/2)]`
Solution
Let 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)`
= `2xlog4 + (3)/(2)log((x^2 + 5)/(sqrt(2x^3 - 4)))`
= `2xlog4 + (3)/(2)[log(x^2 + 5) - log sqrt((2x^3 - 4))^(1/2)]`
= `2xlog4 + (3)/(2)[log(x^2 + 5) - log sqrt(2x^3 - 4)`
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"[2xlog4 + 3/2log(x^2 + 5) - 3/4log(2x^3 - 4)]`
= `(2log4)"d"/"dx"(x)+ (3)/(2)"d"/"dx"[log(x^2 + 5)] - (3)/(4)"d"/"dx"[log(2x^3 - 4)]`
= `(2log4) xx 1 + 3/2 xx (1)/(x^2 + 5)."d"/"dx"(x^2 + 5) - 3/4 xx (1)/(2x^3 - 4)."d"/"dx"(2x^3 - 4)`
= `2log4 + (3)/(2(x^2 + 5)) xx (2x + 0) - (3)/(4(2x^3 - 4)) xx (2 xx 3x^2 - 0)`
= `2log4 + (3x)/(x^2 + 5) - (9x^2)/(2(2x^3 - 4)`.
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