![]() ![]() The derivative of the square root is given in the Example of Lesson 6. G is x 4 − 2 because that is inside the square root function, which is f. What function is f, that is, what is outside, and what is g, which is inside? We take the derivative of f with respect to g first. That is why we take that derivative first. To decide which function is outside, decide which you would have to evaluate last. We then multiply by the derivative of what is inside. When we take the outside derivative, we do not change what is inside. Note: In ( x 2+ 1) 5, x 2+ 1 is "inside" the 5th power, which is "outside." We take the derivative from outside to inside. Therefore according to the chain rule, the derivative of Times the derivative of g( x) with respect to x." Is equal to the derivative of f( g) with respect to g Then the derivative of f with respect to x In the German mathematician Gottfried Wilhelm Leibniz’s notation, which uses d/ d x in place of D and thus allows differentiation with respect to different variables to be made explicit, the chain rule takes the more memorable “symbolic cancellation” form:ĭ( f( g( x)))/ d x = d f/ d g ∙ d g/ d x."If f is a function of g and g is a function of x, In the example of sin x 2, the rule gives the resultĭ(sin x 2) = Dsin( x 2) ∙ D( x 2) = (cos x 2) ∙ 2 x. ![]() In other words, the first factor on the right, D f( g( x)), indicates that the derivative of f( x) is first found as usual, and then x, wherever it occurs, is replaced by the function g( x). The chain rule states that the derivative D of a composite function is given by a product, as D( f( g( x))) = D f( g( x)) ∙ D g( x). If f( x) and g( x) are two functions, the composite function f( g( x)) is calculated for a value of x by first evaluating g( x) and then evaluating the function f at this value of g( x), thus “chaining” the results together for instance, if f( x) = sin x and g( x) = x 2, then f( g( x)) = sin x 2, while g( f( x)) = (sin x) 2. SpaceNext50 Britannica presents SpaceNext50, From the race to the Moon to space stewardship, we explore a wide range of subjects that feed our curiosity about space!Ĭhain rule, in calculus, basic method for differentiating a composite function.Learn about the major environmental problems facing our planet and what can be done about them! ![]() Saving Earth Britannica Presents Earth’s To-Do List for the 21st Century.Britannica Beyond We’ve created a new place where questions are at the center of learning.100 Women Britannica celebrates the centennial of the Nineteenth Amendment, highlighting suffragists and history-making politicians.COVID-19 Portal While this global health crisis continues to evolve, it can be useful to look to past pandemics to better understand how to respond today. ![]() Student Portal Britannica is the ultimate student resource for key school subjects like history, government, literature, and more.This Time in History In these videos, find out what happened this month (or any month!) in history.#WTFact Videos In #WTFact Britannica shares some of the most bizarre facts we can find.Demystified Videos In Demystified, Britannica has all the answers to your burning questions.Britannica Explains In these videos, Britannica explains a variety of topics and answers frequently asked questions.Britannica Classics Check out these retro videos from Encyclopedia Britannica’s archives. ![]()
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