Differentiating Functions Given in the Form of a Product and in the Form of a Quotient

Understanding Product and Quotient Rule in Differentiation

Get familiar with the product rule, an essential method in calculus that is used for differentiating the product of two functions.
Remember that the product rule states that the derivative of a function which is the product of two other functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.
Take note of the formula, (u * v)' = u' * v + u * v' where u and v are the two functions.
Pay attention to the quotient rule, a fundamental method in calculus for differentiating the quotient of two functions.
Understand that the quotient rule states that the derivative of a function which is the quotient of two other functions is the derivative of the top function times the bottom function minus the top function times the derivative of the bottom function, all divided by the square of the bottom function.
Keep in mind the formula, (u / v)' = (u' * v - u * v')/v^2 where u and v are the two functions.

Review how to identify when a function necessitates the use of the product rule or quotient rule for differentiation.
Practice the application of the product rule and quotient rule with a variety of functions, including polynomials, exponential functions, logarithmic functions, and trigonometric functions.
Recognise situations where it may be required to utilise both the product rule and the quotient rule in the same problem.

Master the use of both the product rule and the quotient rule when applied to functions involving the trigonometric functions.
Know the derivatives of sin(x), cos(x) and tan(x) and understand how to incorporate these whilst using the product and quotient rule.

Work through various problems and examples, from straightforward to complicated ones, to further reinforce your learning and skill in applying the product and quotient rules.
Utilise the solutions provided to grasp the step-by-step application of the product or quotient rule.
Understand the typical mistakes that may occur when using the product and quotient rules, and how to avoid them.

Recognise the use of the product rule and quotient rule in solving practical problems in fields such as physics and engineering, particularly those problems involving rates of change.
Understand the relationship between the product and quotient rules and their integrals, i.e., the product and quotient rules are the differentiation analogues of the integration by parts and the integration of rational functions, respectively.