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# Difference Quotient Calculator

Difference Quotient Calculator calculates the difference quotient of the given function. The difference quotient formula gives the slope of a secant line that is drawn to a curve and passes through any two points of that curve.

## What is a Difference Quotient Calculator?

Difference Quotient Calculator is an online tool that helps to compute the difference quotient of the given function. The difference quotient formula is part of the definition of a derivative. To use this * difference quotient calculator*, enter the function in the given input box.

### Difference Quotient Calculator

## How to Use Difference Quotient Calculator?

Please follow the steps given below to find the difference quotient of the given function using the online difference quotient calculator:

**Step 1:**Go to Cuemath’s online difference quotient calculator.**Step 2:**Enter the function in the given input box of the difference quotient calculator.**Step 3:**Click on the**"Calculate"**button to calculate the difference quotient of the given function.**Step 4:**Click on the**"Reset"**button to clear the field and enter new values.

## How Does Difference Quotient Calculator Work?

Suppose we have a function f(x). Say we have a secant line that intersects the curve at two points given by the coordinates (x, f(x)) and (x + h, f(x + h)). Then we calculate the slope of the secant line as follows:

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

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

The equation given above is also the formula for the difference quotient. If h → 0, then the secant becomes the tangent to the curve. In other words, if h → 0 then the difference quotient formula will give the derivative of the given function.

The steps to find the difference quotient for f(x) are given as follows:

- Substitute f(x) with f(x + h).
- Now subtract f(x) from f(x + h); f(x + h) - f(x)
- Open the brackets and simplify this expression.
- Divide this reduced expression by h. The value so obtained will be the difference quotient of the given function.

## Solved Examples on Difference Quotient

**Example 1:** Find the difference quotient for f(x) = 2x + 5 and verify it using the difference quotient calculator.

**Solution**:

Given: f(x) = 2x + 5

f(x + h) = 2(x + h) + 5

= 2x + 2h + 5

Difference quotient = [ f(x + h) - f(x) ] / h

= [2x + 2h + 5 - 2x - 5] / h

= 2h / h

= 2

**Example 2:** Find the difference quotient for f(x) = (x^{2} - 1) / 3 and verify it using the difference quotient calculator.

**Solution**:

Given: f(x) = (x^{2} - 1) / 3

f(x + h) = [(x + h)^{2} - 1] / 3

= [x^{2} + 2xh + h^{2} - 1] / 3

Difference quotient = [ f(x + h) - f(x) ] / h

= [(x^{2} + 2xh + h^{2} - 1) - (x^{2} - 1) ] / 3h

= (h + 2x)/ 3

Now, try the difference quotient calculator and find the difference quotient for:

- f(x) = 5x
^{2}+ 7 - f(x) = - 4x + 9

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