Postby sdbielz » Mon Mar 01, 2010 1:31 pm
I don't know what is meant by the 'four step process', perhaps the four steps I used to get the answer is what they want. Here's how I would do the problem...
f'(x)=2+(3/x) becomes f(x)=2+3^-1, so then f'(x)= -3^-2 or f'(x)=-3/x^2
or else the 'four step process' might mean to find it the long way, although this requires way more than 4 steps:
{lim as h->0} [f(x+h)-f(x)]/h
-fill in f(x+h) and f(x) to get
{lim as h->0} [(2+3/x+h)-(2+3/x)]/h
-find the common denominator for each parenthetic term in the numerator to get
{lim as h->0} [((2x+2h+3)/(x+h))-((2x+3)/x)]/h
-find common denominator for the entire numerator to get
{lim as h->0} [(x(2x+2h+3)/x(x+h))-((x+h)(2x+3)/x(x+h))]/h
-expand the numerator of the numerator to get
{lim as h->0} [(2x^2+2hx+3x-2x^2-2xh-3x-3h)/x(x+h))]/h
-Simplify the numerator of the numerator to get
{lim as h->0} [-3h/x(x+h)]/h
-copy dot flip (multiply numerator by 1/h) to get
{lim as h->0} -3h/hx(x+h)
-cancell out the h's to get
{lim as h->0} -3/x(x+h)
-set h equal to zero to get
-3/x^2
since that was WAY MORE than 4 steps, you're probably safer using the first example.