document.write( "Question 945600: find the derivative of the following function using the limt defintion
\n" ); document.write( "((x-1)^2)/(x)
\n" ); document.write( "I can know the answer is 2.(x-1)/(x) but I keep getting it wrong when using the defintion. could you show me step-by-step how to get that final answer? \n" ); document.write( "
Algebra.Com's Answer #576789 by rothauserc(4718)\"\" \"About 
You can put this solution on YOUR website!
the limit definition states,
\n" ); document.write( "f'(a) = limit as x approaches a of (f(x) - f(a)) / (x-a) = limit as h approaches 0 of (f(a+h) - f(a)) / h
\n" ); document.write( "we are given f(x) = ((x-1)^2)/(x), therefore
\n" ); document.write( "(f(a+h) - f(h)) / h = ((a+h-1)^2/(a+h) - (a-1)^2/a) / h
\n" ); document.write( "= (a(a+h-1)^2 - (a+h)(a-1)^2) / (a(a+h)h)
\n" ); document.write( "= (a(a^2+2ah+h^2-2a-2h+1) - ((a+h)(a^2-2a+1)) / (a^2+ah)h)
\n" ); document.write( "= (a^3+2ha^2+ah^2-2a^2-2ah+a-a^3+2a^2-a-ha^2+2ah-h) / (a^2+ah)h)
\n" ); document.write( "= (ha^2 -h) / (a^2+ah)h)
\n" ); document.write( "= h(a^2-1) / (a^2+ah)h)
\n" ); document.write( "= (a^2-1) / (a^2+ah)
\n" ); document.write( "now the limit as h approaches 0 = (a^2-1) / a^2
\n" ); document.write( "= (a^2 -1) / a^2
\n" ); document.write( "= 1 - 1/a^2
\n" ); document.write( "= 1 - 1/x^2
\n" ); document.write( " \n" ); document.write( "
\n" );