Simplifying the Difference Quotient for f(x) = 1/x

In summary, a difference quotient is a mathematical expression used to find the average rate of change of a function over a specific interval. It is calculated by plugging in the values of f(x+h) and f(x) into the formula (f(x+h)-f(x))/h and simplifying the expression. The significance of the difference quotient lies in its ability to approximate the slope of a curve at a specific point and find the derivative of a function. It can be used for any differentiable function, and is related to the concept of limits as a type of one-sided limit. This relationship is important in calculus for finding the slope of a curve at a specific point.
  • #1
tmt1
234
0
I have a problem:

f(x) = 1/x,

[f(x) - f(a)] / x- a

I am wondering how to approach this problem.

I have so far.

(1/x - 1/a) / (x-a)

([a-x] / xa) / (x-a)

How would I simplify this?

By the way, the answer is

-1 / ax
 
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  • #2
Here's a re-write of your own approach using $\LaTeX$

$$\frac{f(x) - f(a)}{x - a} = \frac{\frac1{x} - \frac1{a}}{x - a} = \frac{\frac{a-x}{ax}}{x - a} = \frac{a - x}{ax(x-a)}$$

Can you use the fact $a - x = -(x-a)$?
 
Last edited:

FAQ: Simplifying the Difference Quotient for f(x) = 1/x

What is a difference quotient?

A difference quotient is a mathematical expression used to find the average rate of change of a function over a specific interval. It is represented by the formula (f(x+h)-f(x))/h, where h represents the change in the input variable and f(x) represents the function.

How is a difference quotient calculated?

To calculate a difference quotient, you need to plug in the values of f(x+h) and f(x) into the formula (f(x+h)-f(x))/h. Then, simplify the expression to get the average rate of change of the function over the given interval.

What is the significance of the difference quotient?

The difference quotient is used to approximate the slope of a curve at a specific point. It helps in understanding the average rate of change of a function and can be used to find the derivative of a function.

Can the difference quotient be used for any type of function?

Yes, the difference quotient can be used for any differentiable function. This means that the function must have a well-defined derivative at the given point.

How is the difference quotient related to the concept of limits?

The difference quotient is a type of limit, specifically a one-sided limit. As h approaches 0, the difference quotient approaches the derivative of the function at the given point. This relationship is important in calculus and helps in finding the slope of a curve at a specific point.

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