![]() ![]() time curve at a specific time of the day. This is equivalent to finding the slope of the energy vs. Plugging a specific value for the time variable into the function for power would then yield the power consumption of our home during a specified hour. If we were to take the derivative of this function, we would get a function of power with respect to time. If we were to record the average energy usage of our home (in kWh) for each hour of a day, we would likely get a composite function characterizing energy consumption with respect to time. Let's say that we are interested in installing auxiliary power sources for our home such as a solar panel array or a generator. Now that we have a better understanding of energy and power, let's go back to our example. If we consumed 1000 Joules of energy per second, we would have 1,000 W (1 kW) of power consumption. Power is the amount of energy consumed per unit time. A second way, using Leibnizs notation for the derivative is: If y is a function of u ( x), then d y d x d y d u d u d x. Perhaps the one you see most commonly in introductory calculus text books is this: The derivative of f ( g ( x)) is given by f ( g ( x)) ( g ( x)). So, if we were to run a single 1000 W microwave oven for 1 hour, we would consume 1 kWh of energy (1,000 W = 1 kW). A watt, commonly notated with a "W", is a unit of measurement for power. So what does the chain rule say There are a few ways of writing it. A kWh is a unit of energy (energy is the ability to do work). When reading the electrical meter for a home, we see that we are given the unit of kWh, or kilowatt-hour. ![]() Solar Panel Arrayįirst, a little background information on energy and power. ![]() Having the ability to take the derivative of a function composed of other functions is valuable, especially when it saves us from having to expand functions like ( x 2-3 x+10) 9 before taking the derivative.īut how can the Chain Rule help us when we are not in the classroom? To answer this question, let's consider power consumption in our home at specific times, and in turn, using this information to evaluate the infrastructure requirements for a solar panel array or a generator. 153 Sometimes it is helpful to rewrite the function so you can see the outside and inside functions better.When we are in our calculus classes, the Chain Rule is extremely helpful when working with composite functions. ![]() 153 derivative of the outside function derivative of the inside functionġ5 # 15, pg. derivative of outside function derivative of inside functionġ2 The chain rule enables us to find the slope of parametrically defined curves:ĭivide both sides by The slope of a parametrized curve is given by:ġ3 Example: These are the equations for an ellipse.ġ4 # 13, pg. Every derivative problem could be thought of as a chain-rule problem: The derivative of x is one. The most common mistake on the chapter 3 test is to forget to use the chain rule. After the chain rule is applied to find the derivative of a function ( ). The following intuitive examples may help to motivate why the chain rule is based on a product of two rates of change. (That’s what makes the “chain” in the “chain rule”!)ġ0 Derivative formulas include the chain rule!ġ1 Every derivative problem could be thought of as a chain-rule problem: While the chain rule is not rigorously proved here (see Appendix E.4), we hope to extend our intuition about where it comes from. 150Ĩ Another example: It looks like we need to use the chain rule again! derivative of the outside function derivative of the inside functionĩ Another example: The chain rule can be used more than once. …then the inside function This is called the “Outside-Inside” Rule. Note that because two functions, g and h, make up the composite. For example, if a composite function f ( x) is defined as. What is needed is a way to combine derivative rules to evaluate more complicated functions.ĥ and one more: This pattern is called the chain rule.ħ Here is a faster way to find the derivative:ĭifferentiate the outside function. 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. Of course, many of the functions that we will encounter are not so simple. Presentation on theme: "AP Calculus AB 4.1 The Chain Rule."- Presentation transcript:Ģ We now have a pretty good list of “shortcuts” to find derivatives of simple functions. ![]()
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