![]() Which represents the slope of the tangent line at the point (−1,−32). A technique that is sometimes suggested for differentiating composite functions is to work from the “outside to the inside” functions to establish a sequence for each of the derivatives that must be taken.Įxample 1: Find f′( x) if f( x) = (3x 2 + 5x − 2) 8.Įxample 2: Find f′( x) if f( x) = tan (sec x).Įxample 5: Find the slope of the tangent line to a curve y = ( x 2 − 3) 5 at the point (−1, −32).īecause the slope of the tangent line to a curve is the derivative, you find that Here, three functions- m, n, and p-make up the composition function r hence, you have to consider the derivatives m′, n′, and p′ in differentiating r( x). If a composite function r( x) is defined as Note that because two functions, g and h, make up the composite function f, you have to consider the derivatives g′ and h′ in differentiating f( x). For example, if a composite function f( x) is defined as The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary. Volumes of Solids with Known Cross Sections.Second Derivative Test for Local Extrema.First Derivative Test for Local Extrema.Differentiation of Exponential and Logarithmic Functions.Differentiation of Inverse Trigonometric Functions.Limits Involving Trigonometric Functions.What do you notice about the relationship between the slopes of the tangents to f and g, and the slope of the tangent to the composite function? Move the slider and see if your conjecture holds. On this same example, also notice that the slope of the tangent line is shown in the upper right corner of each graph. Click the "Equalize Axes" button, then move the slider and notice how the colored line segments illustrate the composition process. Here, the x input is the original red input to f, and the output is the blue output of g. The right hand graph shows the composition. The graph of g also shows a blue vertical line segment which represents the y output of g. Since f is the inner function of the composition, the green y output of f becomes the x input for g, and you can see a green horizontal line on the middle graph showing the x input for g (these two green line segments should be the same length, if you have equalized axes). The length of the red line represent the x input to f, and the green vertical line represents the y output of f. On the graph of f (on the left) you will see a red square (which is draggable) and a red line. The graphs are shown in purple, and each has a tangent line (hard to see for f, because f is also a line). Special cases: Two special cases of the chain rule come up so often, it is worth explicitly noting them. Once you have a grasp of the basic idea behind the chain rule. It is safest to use separate variable for the two functions. The chain rule is a formula to calculate the derivative of a composition of functions. This makes the chain rule a powerful tool for computing derivatives of very complex functions, which can be broken up into compositions of simpler functions. ![]() ![]() See About the calculus applets for operating instructions. When we use the chain rule we need to remember that the input for the second function is the output from the first function. The chain rule as a computational procedure As the last example demonstrates, the chain rule can be applied multiple times in a single derivation. This device cannot display Java animations.
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