The Difference Quotient What is the Difference Quotient

The Difference Quotient

What is the Difference Quotient? Consider a function f(x). Let P be an any point (x, f(x)) on the function f(x), h be a non-zero real number, and Q be the point (x+h, f(x+h)). Q P (x, f(x)) y = f(x) (x+h, f(x+h))

Using the Difference Quotient A Linear Function f(x) = 2 x + 6 Substitute x+h for x in f(x) and subtract f(x). Divide by h. Distribute to clear parentheses. Combine like terms. Simplify the fraction. = 2.

A Quadratic Function • Substitute x+h for x in f(x) and subtract f(x). Divide by h. FOIL to simplify and distribute to clear parentheses. Distribute to clear remaining parentheses. Combine like terms. Factor numerator. Simplify the fraction.

A Square Root Function • Substitute x+h for x in f(x) and subtract f(x). Divide by h. Distribute to clear parenthesis. Combine like terms. Multiply numerator and denominator by the conjugate (the sum’s same terms, but with a flipped sign: i. e. addition to subtraction, or subtraction to addition) of the numerator. Multiply to simplify. Use difference of two squares formula (a+b)(a-b) = a 2 – b 2. Leave denominator in factored form. Simplify. Combine like terms. Simplify the fraction.

A Cubic Function f(x) = x 3 Substitute x+h for x in f(x) and subtract f(x). Divide by h. Simplify. Combine like terms. Factor numerator. Simplify the fraction. = 3 x 2 + 3 xh+ h 2.

Instantaneous Rate of Change The Derivative •

Finding the Derivative: A Reexamination of Previous Examples

f(x) = 2 x+6 (Linear Function) Use the definition of a derivative, (slide #8). Substitute the difference quotient evaluated on slide #4. To evaluate the limit, substitute 0 for every h in the expression. Since there is no h, the expression remains unchanged. In other words, the limit of a constant is the constant. = 2.

Use the definition of a derivative from slide #8. Substitute the difference quotient evaluated on slide #5. Substitute zero for h to evaluate the limit. Simplify.

Use the definition of a derivative. Substitute the difference quotient evaluated on slide #6. Substitute zero for h to evaluate the limit. Combine like terms to simplify.

Use the definition of a derivative. Substitute the difference quotient evaluated on slide #7. Substitute zero for h to evaluate limit. Simplify.

Practice problems •

Answers •

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