We can take a more formal look at the derivative of h ( x ) = sin ( x 3 ) h ( x ) = sin ( x 3 ) by setting up the limit that would give us the derivative at a specific value a a in the domain of h ( x ) = sin ( x 3 ). In addition, the change in x 3 x 3 forcing a change in sin ( x 3 ) sin ( x 3 ) suggests that the derivative of sin ( u ) sin ( u ) with respect to u, u, where u = x 3, u = x 3, is also part of the final derivative. First of all, a change in x x forcing a change in x 3 x 3 suggests that somehow the derivative of x 3 x 3 is involved. This chain reaction gives us hints as to what is involved in computing the derivative of sin ( x 3 ). We can think of this event as a chain reaction: As x x changes, x 3 x 3 changes, which leads to a change in sin ( x 3 ). Consequently, we want to know how sin ( x 3 ) sin ( x 3 ) changes as x x changes. We can think of the derivative of this function with respect to x as the rate of change of sin ( x 3 ) sin ( x 3 ) relative to the change in x. To put this rule into context, let’s take a look at an example: h ( x ) = sin ( x 3 ). Instead, we use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. However, using all of those techniques to break down a function into simpler parts that we are able to differentiate can get cumbersome. 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. In this section, we study the rule for finding the derivative of the composition of two or more functions. However, these techniques do not allow us to differentiate compositions of functions, such as h ( x ) = sin ( x 3 ) h ( x ) = sin ( x 3 ) or k ( x ) = 3 x 2 + 1. ) as well as sums, differences, products, quotients, and constant multiples of these functions. We have seen the techniques for differentiating basic functions ( x n, sin x, cos x, etc. 3.6.5 Describe the proof of the chain rule.3.6.4 Recognize the chain rule for a composition of three or more functions.3.6.3 Apply the chain rule and the product/quotient rules correctly in combination when both are necessary.3.6.2 Apply the chain rule together with the power rule.3.6.1 State the chain rule for the composition of two functions.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. 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. 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. When we are in our calculus classes, the Chain Rule is extremely helpful when working with composite functions.
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