By the definition of the derivative,

#[{f(x)}/{g(x)}]'=lim_{h to 0}{f(x+h)/g(x+h)-f(x)/g(x)}/{h}#

by taking the common denominator,

#=lim_{h to 0}{{f(x+h)g(x)-f(x)g(x+h)}/{g(x+h)g(x)}}/h#

by switching the order of divisions,

#=lim_{h to 0}{{f(x+h)g(x)-f(x)g(x+h)}/h}/{g(x+h)g(x)}#

by subtracting and adding #f(x)g(x)# in the numerator,

#=lim_{h to 0}{{f(x+h)g(x)-f(x)g(x)-f(x)g(x+h)+f(x)g(x)}/h}/{g(x+h)g(x)}#

by factoring #g(x)# out of the first two terms and #-f(x)# out of the last two terms,

#=lim_{h to 0}{{f(x+h)-f(x)}/h g(x)-f(x){g(x+h)-g(x)}/h}/{g(x+h)g(x)}#

by the definitions of #f'(x)# and #g'(x)#,

#={f'(x)g(x)-f(x)g'(x)}/{[g(x)]^2}#

I hope that this was helpful.