CalculusAdditional / Further Maths

The quotient rule - notation

Differentiate a quotient using and .

  • Write , with . Here and .
  • .
  • Keep the order: second function times derivative of first, minus first function times derivative of second.
  • Divide the entire numerator by the square of the denominator; retain its original restrictions.

Example 1

Differentiate with respect to . Work on .

Numerator.Denominator.Derivative of the first function.Derivative of the second function.Use the same factor order as the product rule, with a minus sign.Substitute each function and derivative.

Example 2

Differentiate with respect to . Work on .

Numerator.Denominator.Derivative of the first function.Derivative of the second function.Use the same factor order as the product rule, with a minus sign.Substitute each function and derivative.

Your turn

Question 1

Differentiate with respect to . Work on .

Check answer
Numerator.Denominator.Derivative of the first function.Derivative of the second function.Use the same factor order as the product rule, with a minus sign.Substitute each function and derivative.
Question 2

Differentiate with respect to . Work on .

Check answer
Numerator.Denominator.Derivative of the first function.Derivative of the second function.Use the same factor order as the product rule, with a minus sign.Substitute each function and derivative.
Question 3

Differentiate with respect to . Work on .

Check answer
Numerator.Denominator.Derivative of the first function.Derivative of the second function.Use the same factor order as the product rule, with a minus sign.Substitute each function and derivative.
Question 4

Differentiate with respect to . Work on .

Check answer
Numerator.Denominator.Derivative of the first function.Derivative of the second function.Use the same factor order as the product rule, with a minus sign.Substitute each function and derivative.
Question 5

Differentiate with respect to . Work on .

Check answer
Numerator.Denominator.Derivative of the first function.Derivative of the second function.Use the same factor order as the product rule, with a minus sign.Substitute each function and derivative.
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The quotient rule - f(x), g(x) notationThe product rule - u, v notationThe second derivative