Difference quotient::: how do you solve 1-x^2 for the difference quotient!!!!!!!!!!!please helPP! !~?

Question

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

please show me the steps!!! THANKS!!

Answer 1

I can certainly understand your confusion. The function you're working with is f(x) = 1 - x^2. In this case, the "x" is just a placeholder, kind of like a box we want to fill. I could just as well write f(?) = 1 - ?^2. In the difference quotient, that "box" is being "filled" once by "x+h" and once by "x". So:

(f(x+h) - f(x))/h
= ((1 - (x+h)^2) - (1-x^2))/h
= ((1 - (x^2 + 2xh + h^2)) - (1 - x^2))/h
= (1 - x^2 - 2xh - h^2 - 1 + x^2)/h
= (-2xh - h^2)/h
= -2x - h

By far the most common mistake is using f(x) + h instead of f(x+h). For instance, here f(x+h) = 1 - (x+h)^2, but f(x) + h = 1 - x^2 + h. If you make this mistake, you'll always come out with a "difference quotient" of 1:

(f(x) + h - f(x))/h = h/h = 1

Answer 2

[1+(x+h)^2 -(1+x^2)]/h = 2 x +h

Answer 3

f(x+h)=1-(x+h)^2=-2xh-h^2
f(x)=1-x^2

[f(x+h)-f(x)]/h=(-2xh-h^2)/h=-2x-h