What could be simpler? The reason for the simple form of the chain rule for linear functions is that the derivatives were constants, independent of the value of the inputs to the functions. The following applet illustrates the chain rule for linear functions. If you understand how these points are calculated, then you'll correctly compute the chain rule even for nonlinear functions. Since almost every case where we want to use the chain rule will involve nonlinear functions, evaluating the derivatives at the right points is a crucial step.
The conventions are nearly identical to those one uses for cobwebbing the solution to function iteration. The chain rule for linear functions. In this case of linear functions, the chain rule is quite simple since the slopes are independent of the points where they are evaluated.
You can zoom in, zoom out, or pan the axes by clicking the corresponding buttons. More information about applet. If you understand the chain rule for linear functions, including where to evaluate the derivative, there isn't much more to understanding the chain rule for nonlinear functions. The only difference is that the tangent line to the graph of a nonlinear function does depend on the point at which you calculate the tangent line. The following applet uses the same conventions as the above applet.
It just is a lot messier than the above linear version because we need to plot the tangent lines which depend on the points where we evaluate the functions. The chain rule as multiplying slopes. The slopes are shown near the green symbols, and the tangent lines are shown as thin lines of the same color as the function graphs.
This page focused exclusively on the idea of the chain rule. Of course, knowing the general idea and accurately using the chain rule are two different things. If you are new to the chain rule, check out some simple chain rule examples. If we look at this situation in general terms, we can generate a formula, but we do not need to remember it, as we can simply apply the chain rule multiple times.
Notice that the derivative of the composition of three functions has three parts. Similarly, the derivative of the composition of four functions has four parts, and so on. Also, remember, we can always work from the outside in, taking one derivative at a time. A particle moves along a coordinate axis. At this point, we present a very informal proof of the chain rule.
This notation for the chain rule is used heavily in physics applications. Using the quotient rule,. It is important to remember that, when using the Leibniz form of the chain rule, the final answer must be expressed entirely in terms of the original variable given in the problem.
Learning Objectives State the chain rule for the composition of two functions. Apply the chain rule together with the power rule. Recognize the chain rule for a composition of three or more functions. Describe the proof of the chain rule. Deriving the Chain Rule When we have a function that is a composition of two or more functions, we could use all of the techniques we have already learned to differentiate it. The Chain and Power Rules Combined We can now apply the chain rule to composite functions, but note that we often need to use it with other rules.
Solution Because we are finding an equation of a line, we need a point. Hint Use the preceding example as a guide. Combining the Chain Rule with Other Rules Now that we can combine the chain rule and the power rule, we examine how to combine the chain rule with the other rules we have learned.
Solution First apply the product rule, then apply the chain rule to each term of the product. Hint Start out by applying the quotient rule. Composites of Three or More Functions We can now combine the chain rule with other rules for differentiating functions, but when we are differentiating the composition of three or more functions, we need to apply the chain rule more than once. Then apply the chain rule several times.
Proof of Chain Rule At this point, we present a very informal proof of the chain rule. Answer Finally, we put it all together. Key Concepts The chain rule allows us to differentiate compositions of two or more functions.
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