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Question
Differentiate the following w.r.t.x: `log[(ex^2(5 - 4x)^(3/2))/root(3)(7 - 6x)]`
Solution
Let y = `log[(e^(x^2)(5 - 4x)^(3/2))/root(3)(7 - 6x)]`
Using
log(A.B) = logA + logB
y = `loge^(x^2) + log(((5 - 4x)^(3/2))/root(3)(7 - 6x))`
= `loge^(x^2) + log(5 - 4x)^(3/2) - log(root(3)(7 - 6x))`
= `x^2loge + 3/2log(5 - 4x) - log(7 - 6x)^(1/3)`
= `x^2 + 3/2log(5 - 4x) - 1/3log(7 - 6x)`
Now,
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"x^2 + 3/2"d"/"dx"log(5 - 4x) - 1/3"d"/"dx"log(7 - 6x)`
= `2x + (3)/(2)(1)/(5 - 4x)(-4) - (1)/(3)(1)/((7 - 6x))x(-6)`
= `2x - (6)/((5 - 4x)) + (2)/((7 - 6x)`
`2x - (6)/(5 - 4x) + (2)/(7 - 6x)`.
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