You can
put this solution on YOUR website! I don't understand how to find the
'difference quotient'. Could you please help me?
The problem is:
f(x) = x² + 1
Learn the difference quotient formula:
f(x + h) - f(x)
difference quotient formula = —————————————————
h
First find f(x + h). Start with:
f(x) = x² + 1
Replace both x's by (x + h) :
f(x + h) = (x + h)² + 1
Simplify the right side. To square a binomial
multiply it by itself:
f(x + h) = (x + h)(x + h) + 1
Use FOIL:
f(x + h) = x² + hx + hx + h² + 1
Combine like terms:
f(x + h) = x² + 2hx + h² + 1
Now substitute the following two into
the numerator of the difference quotient formula:
f(x + h) = (x² + 2hx + h² + 1)
f(x) = (x² + 1)
f(x + h) - f(x)
difference quotient formula = —————————————————
h
Substituting into that numerator:
(x² + 2hx + h² + 1) - (x² + 1)
difference quotient = ————————————————————————————————
h
Remove parentheses in the numerator:
x² + 2hx + h² + 1 - x² - 1
difference quotient = ————————————————————————————————
h
The x² and the -x² cancel. The +1 and the -1 cancel,
so we have:
2hx + h²
difference quotient = —————————
h
Factor h out of the numerator:
h(2x + h)
difference quotient = ———————————
h
Cancel the h's
1
h(2x + h)
difference quotient = ———————————
h
1
difference quotient = 2x + h
Edwin
