Find the simplified difference quotient for each function...

Key Concepts

Difference Quotient

The difference quotient is a fundamental concept in calculus that represents the average rate of change of a function over a small interval. It forms the basis for the derivative, as taking the limit of the difference quotient as the interval approaches zero yields the instantaneous rate of change of the function. It is generally expressed as (f(x+h) ? f(x))/h.

Binomial Expansion

The binomial expansion is used to expand expressions raised to a power, such as (x+h)^n. In the context of the difference quotient, it allows us to conveniently expand the function values at (x+h) so that terms can be combined and simplified. This is particularly useful when dealing with polynomial functions, where the binomial theorem provides each term in the expansion with its corresponding coefficient.

Algebraic Simplification

Algebraic simplification involves the process of reducing expressions to their simplest form, typically by canceling common factors, combining like terms, and factoring. In the difference quotient, after expanding the function, simplifying the expression is essential to isolate h and ultimately evaluate the limit if needed.

Transcript

00:01 This problem wants the simplified difference quotient for x the power 4.

00:05 So we're going to plug this into f of x plus h.

00:09 So if it's x the power 4, that would be x plus h to the power 4 minus f of x, which would be x the power 4, all over h.

00:19 So we're going to take this down here and simplify it.

00:22 So x plus h to the power of 4 would be x plus h times itself 4 times.

00:28 So this is going to get a little bit messy.

00:33 So i'm going to multiply these two together, these two together, and then the whole thing together.

00:39 So we're foiling here.

00:42 So what are we going to get? we're going to get x squared plus 2xh plus h squared.

00:49 And you're going to get the same thing for the other one.

00:51 X squared plus 2xh plus h squared.

00:56 Now what you need to do when you're multiplying trinomel.

00:58 Is distribute each of these terms across and you're multiplying each of those terms essentially by each other.

01:08 So i'm going to start with the x squared.

01:10 So x squared times x squared is an x to the power four plus two x squared times x h would be x cubed h plus x squared h squared.

01:22 So i'm done with that x squared.

01:24 I've distributed it.

01:25 So now i'm going to distribute the two x -h.

01:27 So i'll get two x -h.

01:28 So i'll get two x cubed h for distributing to that x squared plus four two times two is four x squared h squared and the last one term there would be plus two two x h cubed now i'm done distributing this one now i have to distribute the x or say the h squared so i'm going to add it to a row beneath i'm going to get x squared h squared and then i'll get two xh cubed and then i'll get plus h to the power of four...

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