Given y = ln((2x+3)/(7x^3 +1)). Find dy/dx
y = ln(2x+3 / 7x^3 +1)
d/dx(2x+3 / 7x^3 + 1) by quotient rule which is(v.du/dx - u.dv/dx) / v^2 where u=2x+3 and v=7x^3 +1 gives (-27x^3 -63x^2 +2) / (7x^3 +1)^2
so d/dx(ln(2x+3 / 7x^3 +1) = ( (-27x^3 -63x^2 +2) / (7x^3 +1)^2 ) / (2x+3)/(7x^3 +1)
= ((-27x^3 -63x^2 +2) / (7x^3 +1) ) / (2x+3)
= (-27x^3 -63x^2 +2) / (7x^3 +1).(2x+3)
which is the final solution, since it cannot be simplified further.
SB
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