![]() Take a Tour and find out how a membership can take the struggle out of learning math. But these chain rule/product rule problems are going to require power rule, too. Still wondering if CalcWorkshop is right for you? In this lesson, we want to focus on using chain rule with product rule. If y g(x)+, then we can write y f(u) u where u g(x). Get access to all the courses and over 450 HD videos with your subscription The product rule is applied to functions that are the product of two terms, which both depend on x, for example, y (x - 3)(2x2 - 1). The chain rule now joins the sum, constant multiple, product, and quotient rules in our collection of techniques for finding the derivative of a function. The power rule combined with the Chain Rule This is a special case of the Chain Rule, where the outer function f is a power function. Well, it works in the first stage, i. The function is ( 625 x 2) 1 / 2, the composition of f ( x) x 1 / 2 and g ( x) 625 x 2. Let’s get to it! Video Tutorial w/ Full Lesson & Detailed Examples (Video) This is a quotient with a constant numerator, so we could use the quotient rule, but it is simpler to use the chain rule. So, throughout this lesson, we will work through numerous examples of the chain rule, combining our previous differentiation rules such as the power rule, product rule, and quotient rule, so that you will become a chain-rule master! ![]() ![]() In fact, we will come to see that the chain rule’s helpfulness extends beyond polynomial functions but is pivotal in how we differentiate: Thanks to the chain rule, we can quickly and easily find the derivative of composite functions - and it’s actually considered one of the most useful differentiation rules in all of calculus. Good grief! That would have been painful. Without it, we would have had to multiply the polynomial you see in blue by itself 10 times, simplify, and then use the power rule to find the derivative! 1 Form the two possible compositions of f ( x) x and g ( x) 625 x 2 and compute the derivatives. ![]() Next, we multiplied by the derivative of the inside function, and lastly, we simplified. In general, if f ( x) and g ( x) are functions, we can compute the derivatives of f ( g ( x)) and g ( f ( x)) in terms of f ( x) and g ( x). See, all we did was first take the derivative of the outside function (parentheses), keeping the inside as is. ![]()
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