![]() Step 1: Differentiate the outer function. However, the technique can be applied to a wide variety of functions with any outer exponential function (like x 32 or x 99. This section shows how to differentiate the function y = 3x + 1 2 using the chain rule. The more times you apply the chain rule to different problems, the easier it becomes to recognize how to apply the rule. That’s it! Chain rule examples: Exponential Functionsĭifferentiating using the chain rule usually involves a little intuition. Step 4: Multiply Step 3 by the outer function’s derivative. The derivative of x 4 – 37 is 4x (4-1) – 0, which is also 4x 3. Step 3: Differentiate the inner function. Step 2:Differentiate the outer function first. The outer function is √, which is also the same as the rational exponent ½. The inner function is the one inside the parentheses: x 4 -37. Step 1: Identify the inner and outer functions.įor an example, let the composite function be y = √(x 4 – 37). That isn’t much help, unless you’re already very familiar with it. Multivariable Chain Rule (Opens in New Window). ![]() If your derivative has an inner and an outer function, you can use the chain rule. The solution? Take a look at the following examples of the chain rule, which illustrate the different ways inner functions and outer functions look. The problem is recognizing those functions that you can differentiate using the rule. The inner function is the one inside the parentheses: x 2 -3. 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.The following chain rule examples show you how to differentiate (find the derivative of) many functions that have an “ inner function” and an “ outer function.” For an example, take the function y = √ (x 2 – 3). 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). See About the calculus applets for operating instructions. ![]() This device cannot display Java animations.
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