difference quotient

Question

Show that the difference quotient f(x)=e^x can be written as e^h/h e

Answer 1

f(x) = e^x

f(x+h) = e^(x+h)

f(x+h) - f(x) = e^(x+h) - e^x = e^x(e^h) -e^x = e^x(e^h-1)

difference quotient = [f(x+h) - f(x)]/h = [e^(x+h) -e^x]/h

= (e^x(e^h) - e^x)/h

=e^x(e^h -1)/h = [e^x)/h])(e^h-1)

= e^x(e^h/h - 1/h)= e^x(1)= e^x, as h appproaches zero

compare the derivative of e^x = e^x

derivative of e^x = slope of the exponential funciton = itself= the difference quotient as h approaches zero