CalculusCourse skill

The product rule - notation

Differentiate a product using and .

  • Write . Here and .
  • .
  • Keep the order: second function times derivative of first, plus first function times derivative of second.

Example 1

Differentiate with respect to .

First factor.Second factor.Derivative of the first function.Derivative of the second function.Use the same factor order as the quotient numerator, with a plus sign.Substitute each function and derivative.
First factor.Second factor.Derivative of the first function.Derivative of the second function.Use the same factor order as the quotient numerator, with a plus sign.Substitute each function and derivative.

Example 2

Differentiate with respect to .

First factor.Second factor.Derivative of the first function.Derivative of the second function.Use the same factor order as the quotient numerator, with a plus sign.Substitute each function and derivative.
First factor.Second factor.Derivative of the first function.Derivative of the second function.Use the same factor order as the quotient numerator, with a plus sign.Substitute each function and derivative.

Your turn

Question 1

Differentiate with respect to .

Check answer
First factor.Second factor.Derivative of the first function.Derivative of the second function.Use the same factor order as the quotient numerator, with a plus sign.Substitute each function and derivative.
First factor.Second factor.Derivative of the first function.Derivative of the second function.Use the same factor order as the quotient numerator, with a plus sign.Substitute each function and derivative.
Question 2

Differentiate with respect to .

Check answer
First factor.Second factor.Derivative of the first function.Derivative of the second function.Use the same factor order as the quotient numerator, with a plus sign.Substitute each function and derivative.
First factor.Second factor.Derivative of the first function.Derivative of the second function.Use the same factor order as the quotient numerator, with a plus sign.Substitute each function and derivative.
Question 3

Differentiate with respect to .

Check answer
First factor.Second factor.Derivative of the first function.Derivative of the second function.Use the same factor order as the quotient numerator, with a plus sign.Substitute each function and derivative.
First factor.Second factor.Derivative of the first function.Derivative of the second function.Use the same factor order as the quotient numerator, with a plus sign.Substitute each function and derivative.
Question 4

Differentiate with respect to .

Check answer
First factor.Second factor.Derivative of the first function.Derivative of the second function.Use the same factor order as the quotient numerator, with a plus sign.Substitute each function and derivative.
First factor.Second factor.Derivative of the first function.Derivative of the second function.Use the same factor order as the quotient numerator, with a plus sign.Substitute each function and derivative.
Question 5

Differentiate with respect to .

Check answer
First factor.Second factor.Derivative of the first function.Derivative of the second function.Use the same factor order as the quotient numerator, with a plus sign.Substitute each function and derivative.
First factor.Second factor.Derivative of the first function.Derivative of the second function.Use the same factor order as the quotient numerator, with a plus sign.Substitute each function and derivative.
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