DIFFERENCE QUOTIENT FORMULA AND EXAMPLES

Examples 1-3 : Find difference quotients for the following functions :

Example 1 :

f(x) = 2x + 5

Solution :

Formula to find difference quotient for f(x).

f(x) = 2x + 5

To evaluate f(x + h), substitute x = x + h in f(x).

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

= 2x + 2h + 5

Difference Quotient of f(x) :

Example 2 :

f(x) = 2x² - 5

Solution :

Formula to find difference quotient for f(x).

f(x) = 2x² - 5

To evaluate f(x + h), substitute x = x + h in f(x).

f(x + h) = 2(x + h)² - 5

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

= 2(x² + xh + xh + h²) - 5

= 2(x² + 2xh + h²) - 5

= 2x² + 4xh + 2h² - 5

Difference Quotient of f(x) :

Example 3 :

f(x) = x² - 3x + 6

Solution :

Formula to find difference quotient for f(x).

f(x) = x² - 3x + 6

To evaluate f(x + h), substitute x = x + h in f(x).

f(x + h) = (x + h)² - 3(x + h) + 6

= (x + h)(x + h) - 3x - 3h + 6

= x² + xh + xh + h² - 3x - 3h + 6

= x² + 2xh + h² - 3x - 3h + 6

Difference Quotient of f(x) :

Example 4 :

f(x) = lnx

Solution :

Formula to find difference quotient for f(x).

f(x) = lnx

To evaluate f(x + h), substitute x = x + h in f(x).

f(x + h) = ln(x + h)

Difference Quotient of f(x) :

Using the quotient rule of logarithms,

Example 5 :

Find the derivative of the following function by applying the limit h --> 0 to the difference quotient formula.

f(x) = 3x² + 7

Solution :

Formula to find difference quotient for f(x).

f(x) = 3x² + 7

To evaluate f(x + h), substitute x = x + h in f(x).

f(x + h) = 3(x + h)² + 7

= 3(x + h)(x + h) + 7

= 3(x² + xh + xh + h²) + 7

= 3(x² + 2xh + h²) + 7

= 3x² + 6xh + 3h² + 7

Difference Quotient of f(x) :

In the above difference quotient of f(x), by taking the limit h --> 0, you will get the derivative of f(x), that is f'(x).

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