It is given that \(y=\tan[\ln(1+2x)]\), for \(x>-0.5\).
Show that \((1+2x)\dfrac{dy}{dx}=2(1+y^2)\).[2]
Hence find the first three non-zero terms of the Maclaurin expansion of \(\tan[\ln(1+2x)]\).[4]
It is given that the three terms found in part (b) are equal to the first three terms in the series expansion of \(2x(1+ax)^b\). Find the exact values of the constants \(a\) and \(b\).[3]