document.write( "Question 1149041: Suppose f(x)=1/x. Write an expression in terms of x and h that represents the average rate of change of f over any interval of length h. [That is, over any interval (x,x+h).] Simplify your answer as much as possible. \n" ); document.write( "

Algebra.Com's Answer #770380 by rothauserc(4718) $\"\"$ $\"About$

You can put this solution on YOUR website!
The problem is asking the first derivative of f(x) using the limit definition of the first derivation, that is,
\n" ); document.write( ":
\n" ); document.write( "f'(x) = limit as h approaches 0 of (f(x+h) -f(x))/h
\n" ); document.write( ":
\n" ); document.write( "We are given f(x) = 1/x
\n" ); document.write( ":
\n" ); document.write( "f(x+h) = 1/(x+h)
\n" ); document.write( ":
\n" ); document.write( "f'(x) = limit as h approaches 0 of ( 1/(x+h) - 1/x) )/h =
\n" ); document.write( ":
\n" ); document.write( "(x - (x+h) ) / ( h * x * (x+h)) =
\n" ); document.write( ":
\n" ); document.write( "-h/(h * x * (x+h)) =
\n" ); document.write( ":
\n" ); document.write( "-1/(x^2 +xh)
\n" ); document.write( ":
\n" ); document.write( "h approaches 0, so we are left with
\n" ); document.write( ":
\n" ); document.write( "f'(x) = -1/x^2
\n" ); document.write( ":
\n" ); document.write( " \n" ); document.write( "

\n" );