**Difference Quotient** is used to calculate the slope of the secant line between two points on the graph of a function, f. Just to review, a function is a line or curve that has only one y value for every x value. The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length h). The limit of the difference quotient (i.e., the derivative) is thus the instantaneous rate of change.

In calculus, the difference quotient is the formula used for finding the derivative which is the difference quotient between two points that are as close as possible which gives the rate of change of a function at a single point. The difference quotient was formulated by Isaac Newton.

**Difference Quotient Calculator**

This formula computes the slope of the secant line through two points on the graph of f. These are the points with x-coordinates x and x + h. The difference quotient is used in the definition of the derivative. First, plug (x + h) into your function wherever you see an x. Once you find f (x + h), you can plug your values into the difference quotient formula and simplify from there. In the third step, you use the subtraction sign to eliminate the parentheses and simplify the difference quotient.

In the formal definition of the difference quotient, you’ll note that the slope we are calculating is for the secant line. A secant line is just any line that passes between two points on a curve. We label these two points as x and (x +h) on our x-axis. Because we are working with a function, these points are labeled as f (x) and f (x + h) on our y-axis, respectively.

In simple terms, the difference quotient helps us find the slope when we are working with a curve. In the case of a curve, we cannot use the traditional formula of:

which is why we must use the difference quotient formula.

**Difference Quotient Formula**

- Plug x + h into the function f and simplify to find f(x + h).
- Now that you have f(x + h), find f(x + h) – f(x) by plugging in f(x + h) and f(x) and simplifying.
- Plug your result from step 2 in for the numerator in the difference quotient and simplify it.

which when taken to the limit as h approaches 0 gives the derivative of the function f. The name of the expression stems from the fact that it is the quotient of the difference of values of the function by the difference of the corresponding values of its argument (the latter is (x+h)-x=h in this case).

**Symmetric Difference Quotients**

In mathematics, the difference quot is formulas that give approximations of the derivative of a function. There are a few different difference quots, and those are the one-sided difference quotients and the symmetric difference quot. They are all related, and one gives a better approximation than the others due to this relationship.

In mathematics, the **symmetric derivative** is an operation generalizing the ordinary derivative. It is defined as:

The expression under the limit is sometimes called the **symmetric difference quotient**. A function is said to be **symmetrically differentiable** at a point x if its symmetric derivative exists at that point.

If a function is differentiable (in the usual sense) at a point, then it is also symmetrically differentiable, but the converse is not true. A well-known counterexample is the absolute value function f(x) = |x|, which is not differentiable at x = 0, but is symmetrically differentiable here with symmetric derivative 0. For differentiable functions, the symmetric difference quotient does provide a better numerical approximation of the derivative than the usual difference quotient.

The symmetric derivative at a given point equals the arithmetic mean of the left and right derivatives at that point if the latter two both exist.

Neither Rolle’s theorem nor the mean value theorem hold for the symmetric derivative; some similar but weaker statements has been proved.

**The modulus function**

For the modulus function, f(x)=|x|, we have, at x=0,

where since h>0 we have |h|=-(-h) . So, we observe that the symmetric derivative of the modulus function exists at x=0, and is equal to zero, even though its ordinary derivative does not exist at that point (due to a “sharp” turn in the curve at x=0).

Note in this example both the left and right derivatives at 0 exist, but they are unequal (one is -1 and the other is 1); their average is 0, as expected.

**Difference Quotient Example**

- The difference quot for the function f(x)=3-x^2-x is:

- The difference quot for the function is:

The difference quot for the function is:

**Evaluate The Difference Quotient**

Some practice problems for you; find the difference quot for each function showing all relevant steps in an organized manner (see examples).

- f(x)=3-7x
- k(t)=7x²+2
- z(x)=π
- s(t)=t³-t-9

**How do you find the quotient?**

Multiply the divisor by a power of 10 to make it a whole number.

Multiply the dividend by the same power of 10. Place the decimal point in the quot.

Divide the dividend by the whole-number divisor to find the quot.