Product and Quotient Rules and Higher-Order Derivatives - University of Texas at El Paso

[Pages:35]Product and Quotient Rules and Higher-Order Derivatives

By Tuesday J. Johnson

Suggested Review Topics

? Algebra skills reviews suggested:

? Multiplying polynomials ? Radicals as rational exponents ? Simplifying rational expressions ? Exponential rules

? Trigonometric skills reviews suggested:

? Derivatives of sine and cosine

Calculus Differentiation

Product and Quotient Rules and Higher-Order Derivatives

The Product Rule

? The product of two differentiable functions and is itself differentiable.

? Moreover, the derivative of is the first function multiplied by the derivative of the second, plus the second function multiplied by the derivative of the first.

? The formula:

= + ()

Examples: Use the product rule to find the derivative.

1. = (6 + 5)(3 - 2)

? Let = 6 + 5 and = 3 - 2. Then we can find the derivatives to be = 6 and = 32.

? Using the product rule we have = 6 + 5 32 + 3 - 2 (6) = 183 + 152 + 63 - 12 = 243 + 152 - 12

Examples: Use the product rule to find the derivative.

1. = 6 + 5 3 - 2

Suppose we didn't use the product rule and we first multiplied the function and simplified to get

= 64 - 12 + 53 - 10 Then we can take the derivative and still get

() = 243 + 152 - 12 Either we multiply to start and take the derivative or take little derivatives and multiply at the end.

Examples: Use the product rule to find the derivative.

2. = sin

Rewrite: = 1/2 sin

Identify:

= 1/2 with

=

1 2

-1/2

and

= sin with = cos .

Product Rule:

=

(1/2)(cos

)

+

(sin

1 )(2

-12)

Rewrite: =

cos + sin

2

Examples: Use the product rule to find the derivative. 3. = -2 + 1 3 - cos

Rewrite: = -2 + -1 1/3 - cos

Identify:

= -2 + -1 = (-2-3 - -2)

= 1/3 - cos = (1 -23 + sin )

3

Product Rule: = -2 + -1

1 -23 + sin +

3

1/3 - cos (-2-3 - -2).

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