$\displaystyle y=\ \text{ln}\left( {\frac{{\sin x-1}}{{3x+2}}} \right)$

Use the Law of Logarithm Rule lg (x/y) = lg x – lg y. to simplify the expression so that it will be easier to differentiate.

$\displaystyle y=\ln (\ \sin x-1)-\ln (\ 3x+2)$

To differentiate y=ln(fx) , use the formula dy/dx= 1/(fx) x differentiate (fx)

$\displaystyle \frac{{dy}}{{dx}}=\frac{1}{{\sin x-1}}\times \cos x-\frac{1}{{3x+2}}\times 3$

$\displaystyle \frac{{dy}}{{dx}}=\frac{{\cos x}}{{\sin x-1}}-\frac{3}{{3x+2}}$

Always try to simplify the expression before you differentiate, if not, you will may have to take many more steps to differentiate the expression.  You need to memorize the formula for differentiation of ln as it is NOT in the formula sheet. See more post regarding the differentiation of ln with cube root

New Elementary Math (E-Math) and Additional Math Group  Tuition Class near Admiralty MRT station.