A common mistake when applying the product rule is to not apply it at all, i.e., to incorrectly apply basic power and constant rules to each function in the product separately. Another mistake is to misapply the rule by adding the derivatives of the individual functions instead of adding the products of the functions and their derivatives. When using the chain rule, a common error is to forget to multiply by the derivative of the inner function. Always ensure that you carefully apply each step of the rule and check your work for errors.

Choosing the appropriate rule for differentiation typically depends on the form of the function. If the function is a product of two functions, use the product rule. If it is a quotient of two functions, use the quotient rule. If it is a composition of two functions, use the chain rule. Sometimes, a function can be simplified to a form where basic differentiation rules can be applied, making it unnecessary to use more complex rules. Always look for opportunities to simplify the function before applying differentiation rules to make the process easier.

Effective practice involves working through a variety of problems that use these rules. Start with simpler functions and gradually work your way up to more complex ones. Ensure that you understand each step of the process and can apply the rules consistently and accurately. Utilise resources such as textbooks, online platforms, and your teachers to find practice problems and solutions. Additionally, try creating your own problems by altering existing ones, ensuring that you understand the underlying principles and not just memorising solutions.

Yes, the product rule can be extended when more than two functions are being multiplied together. Suppose we have three functions, f(x), g(x), and h(x), and we wish to find the derivative of their product. We can use the extended product rule: (fgh)' = f'gh + fg'h + fgh'. Essentially, we take the derivative of each function one at a time, keeping the other functions as they are, and then add all the terms together. This method can be applied to the product of any number of functions by differentiating each function in turn and adding up all the resulting terms.

When dealing with the composition of more than two functions, the chain rule can be extended in a straightforward manner. Suppose we have three functions composed together, say h(x) = f(g(k(x))). To find h'(x), we differentiate f with respect to g, g with respect to k, and k with respect to x, and then multiply these derivatives together: h'(x) = f'(g(k(x))) * g'(k(x)) * k'(x). This method can be extended to any number of composed functions by differentiating each function with respect to the inner function and multiplying all the derivatives together, working from the outermost function to the innermost function.