How do you find the difference quotient of #f#, that is, find #((f(x+h)-f(x))/h)#, #h != 0# for #f(x)=x^2-5x+7#?

1 Answer
Dec 14, 2017

see explanation

Explanation:

If I understand the question correctly, you have to start by substituting #(x+h)# wherever you see #x# in your original function definition:

#(x+h)^2 - 5(x+h) + 7#

multiplying out, you have: #x^2+ 2xh + h^2 - 5(x+h) + 7#

#= x^2 + 2xh + h^2 - 5x - 5h + 7#

So, substitute this as the value of #f(x+h)# in the definition of the difference quotient.

#(f(x+h) - f(x))/h#

# = ((x^2 + 2xh + h^2 - 5x - 5h + 7) - (x^2 - 5x + 7))/h#

...this simplifies to:

# (2xh + h^2 - 5h)/h#

...now, since this is calculus, the inevitable next step is to find the limit of this function as h -> 0. For this, you can't have h in the denominator. But we can do a little factoring kung-fu. The equation above can be re-written:

#(h(2x + h - 5))/h = 2x - 5 + h#

...and the limit of this function, as h approaches 0, is :

#2x - 5#

...which is the derivative of the original function #x^2 - 5x + 7#

GOOD LUCK