Question

1. Differentiate with respect to x:
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Answer 1

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Answer:

a) y' = x^2(3x·ln(6x) +1)

b) y' = 6e^(3x)/(1 -e^(3x))^2

Explanation:

The applicable rules for derivatives include ...

d(u^n)/dx = n·u^(n-1)·du/dx

d(uv)/dx = (du/dx)v +u(dv/dx)

d(e^u)/dx = e^u·du/dx

d(ln(u))/dx = 1/u·du/dx

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(a)

`y=x^3ln((6x))\\\\y'=3x^2ln((6x))+(x^3\cdot6)/(6x)\\\\\boxed{(dy)/(dx)=3x^3ln((6x))+x^2}`

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(b)

`y=(1+e^(3x))/(1-e^(3x))=1+(2)/(1-e^(3x))=1+2(1-e^(3x))^(-1)\\\\y'=-2(1-e^(3x))^(-2) (-3e^(3x))\\\\\boxed{(dy)/(dx)=(6e^(3x))/((1-e^(3x))^2)}`