difference quotient?

Question

find the difference quotient of:
f(x)=-7x^2+9x-6


please show all work so I can start to understand this!!!!
:)

Answer 1

The difference quotient is

[f(x+h) - f(x)]/h

(your book may use some other notation, such as Δx, for the change in x)

f(x+h) = -7(x+h)² + 9(x+h) - 6

Expanding this,

f(x+h) = -7(x² + 2xh + h²) + 9(x+h) - 6 = -7x² - 14xh - 7h² + 9x + 9h - 6

Then

f(x+h) - f(x) = -7x² - 14xh - 7h² + 9x + 9h - 6 - (-7x² + 9x - 6) = -14xh - 7h² + 9h

Note that we can get a factor of h out of this; that's crucial, because we need it to cancel the h in the denominator of the difference quotient. Doing so,

f(x+h) - f(x) = h(-14x - 7h + 9)

And so

[f(x+h) - f(x)]/h = [h(-14x - 7h + 9)]/h = -14x - 7h + 9

Answer 2

f'(x)=(-7(x+h)^2+9(x+h)-6) - (-7x^2+9x-6)]/h
f'(x) = [-7(x^2+2hx+h^2)+9x+9h-6 +7x^2-9x+6]/h
f'(x) = [-7x^2-14hx-7h^2+9x+9h-6 +7x^2-9x+6]/h
f'(x) = [-14hx-7h^2+9h]/h
f'(x) = -14x-7h+9
f'(x) = -14x + 9

Answer 3

Difference qoutient requires two functions