Product rule calculus 29/7/2023 ![]() Please e-mail any correspondence to Duane Koubaīy clicking on the following address About this document. Your comments and suggestions are welcome. Remember your product rule: derivative of the first factor times the second, plus derivative of the second factor times the first. Ĭlick HERE to see a detailed solution to problem 21.Ĭlick HERE to return to the original list of various types of calculus problems. With tangent lines parallel to the line y + x = 12. PROBLEM 21 : Find all points ( x, y) on the graph of.PROBLEM 20 : Find an equation of the line perpendicular to the graph ofĬlick HERE to see a detailed solution to problem 20. ![]() ![]() PROBLEM 19 : Find an equation of the line tangent to the graph ofĬlick HERE to see a detailed solution to problem 19.For what values of x is f'( x) = 0 ?Ĭlick HERE to see a detailed solution to problem 18. Compare it with the ordinary product rule to see the similarities and differences.Ĭlick HERE to see a detailed solution to problem 16.Ĭlick HERE to see a detailed solution to problem 17. Given two differentiable functions, f(x) and g(x), where f'(x) and g'(x) are their respective derivatives, the product rule can be stated as, or using abbreviated notation: The product rule can be expanded for more functions. For what values of x is f'( x) = 0 ?Ĭlick HERE to see a detailed solution to problem 15. The product rule is a formula that is used to find the derivative of the product of two or more functions. For what values of x is f'( x) = 0 ?Ĭlick HERE to see a detailed solution to problem 14. For what values of x is f'( x) = 0 ?Ĭlick HERE to see a detailed solution to problem 13. The following problems require use of the chain rule.Ĭlick HERE to see a detailed solution to problem 7.Ĭlick HERE to see a detailed solution to problem 8.Ĭlick HERE to see a detailed solution to problem 9.Ĭlick HERE to see a detailed solution to problem 10.Ĭlick HERE to see a detailed solution to problem 11.Ĭlick HERE to see a detailed solution to problem 12. In most cases, final answers to the following problems are given in the most simplified form.Ĭlick HERE to see a detailed solution to problem 1.Ĭlick HERE to see a detailed solution to problem 2.Ĭlick HERE to see a detailed solution to problem 3.Ĭlick HERE to see a detailed solution to problem 4.Ĭlick HERE to see a detailed solution to problem 5.Ĭlick HERE to see a detailed solution to problem 6. In the list of problems which follows, most problems are average and a few are somewhat challenging. Each time, differentiate a different function in the product and add the two terms together. The rule follows from the limit definition of derivative and is given by The product rule is a formal rule for differentiating problems where one function is multiplied by another. How a calculus teacher chooses to do this probably says a lot about their pedagogy and educational priorities. In the following discussion and solutions the derivative of a function h( x) will be denoted by or h'( x). At some point in every calculus class, we must discover and prove the product rule for derivatives. ![]() This tells us that when 5 thousand items are produced, the average cost per item is decreasing by $0.The following problems require the use of the product rule. Since the units on \(AC\) are dollars per item, and the units on \(x\) are thousands of items, the units on \(AC'\) are dollars per item per thousands of items. ![]()
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