construct and simplify the difference quotient?

Question

f(x)=x^2+6

Answer 1

f(x) = x^2 + 6

The difference quotient is in this form:

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

Now, all we have to do is algebraically show this. Note that for functions, what appears on the inside goes for every occurrence of x in the function, so we have

{ [(x + h)^2 + 6] - [x^2 + 6] } / h

First, expand the squared binomial.

[x^2 + 2xh + h^2 + 6 - x^2 - 6 ] / h

Now, group like terms. Note that x^2 and -x^2 cancel, as well as the +6 and -6. This leaves us with

[2xh + h^2] / h

Now, factor an h out of the top.

h(2x + h)/h

Now, we can cancel the h; this gives us

2x + h

Answer 2

(3(x+h)^2 + 5(x+h) -3x^2 -5x)/h = (3(x^2+2xh+h^2)+5(x+h)-3x^2 -5x)/h = (3x^2+6xh+3h^2+5x+5h-3x^2 -5x)/h = (6xh+3h^2+5h)/h = 6x+5 +3h. Now understand that as quickly as differientiating, we are taking the shrink as H methods 0. This the term 3h disapears and additionally you're left with 6x+5. it fairly is confirmed by potential of the ability rule.