Nov 20, · In this section we discuss one of the more useful and important differentiation formulas, The Chain Rule. With the chain rule in hand we will be able to differentiate a much wider variety of functions. As you will see throughout the rest of your Calculus courses a great many of derivatives you take will involve the chain rule! The chain rule tells us how to find the derivative of a composite function. Brush up on your knowledge of composite functions, and learn how to apply the chain rule correctly. If you're seeing this message, it means we're having trouble loading external resources on our website.
There is an alternate notation however that while probably not used much in Calculus I is more convenient at this point because it will match up with the notation that we are going to be using in this section.
Here it is. From this point there are still many different possibilities that we can look at. We will be looking at two distinct cases prior to generalizing the whole idea out. This case is analogous to the standard chain rule from Calculus I that we looked at above.
The final step is to then add all this up. Doing this gives. The same result for less work. Note however, that often it will actually be more work to do the substitution first. This is dependent upon the situation, class and instructor however so be careful about not substituting in for without first talking to your instructor.
Now, there is a special case that we should take a quick look at before moving on to the next case. So, not surprisingly, these are very similar to the first case that we looked at. Here is a quick example of this kind of chain rule. We can build up a tree diagram that will give us the chain rule for any situation.
We already know what this is, but it may help to illustrate the tree diagram if we already know the answer. For reference here is the chain rule for this case. We start at the top with the function itself and the branch out from that point.
The first set of chaon is for the variables in the function. We connect each letter with a how to find the right pair of glasses and each line represents a partial derivative as shown.
How do we do those? Here is the first derivative. The issue here is to correctly deal with this derivative. So, the using the product rule gives the following.
This however is exactly what we need to do the two new derivatives we need above. The final topic in this section is a revisiting of implicit differentiation. With these forms of the chain rule implicit differentiation actually becomes a fairly simple process.
Using the chain rule from this section however ccalculus can get a calfulus simple formula for doing this. This will mean using the chain rule on the left side rue the right side will, of course, differentiate to zero.
Here are the results of that. Note as well that in order to simplify the formula we switched back to using the subscript notation for the derivatives.
There we go. It would have taken much longer to do this using the old What is the chain rule in calculus I way of doing this. We can also do ib similar to handle the types of implicit differentiation problems involving partial derivatives like those we saw when we first introduced partial derivatives. Also, the left side will require the chain rule. Here is this derivative. As with the one variable case we switched to the subscripting notation for derivatives to simplify the formulas.
This was one of the functions that we used the old implicit differentiation on back in how to get rid of pimples overnight Partial Derivatives section.
You might want to go back and see the difference between the two. If you go back and compare how to teach pikachu volt tackle answers to those that we found the first time around you will notice that they might appear to be different. However, if you take into account the minus sign that sits in the front of our answers here you will see that they are in fact the same. Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i.
Due to the rue of the mathematics on this site it is best views in landscape mode. If your device is not in rue mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. Example 4 Use a tree diagram to write down the chain rule for the given derivatives.
Here is the tree diagram for this situation. Show Solution This was one of the functions that we used the old implicit differentiation on back in the Partial Derivatives section.
May 31, · Case 1: z = f (x,y) z = f (x, y), x = g(t) x = g (t), y =h(t) y = h (t) and compute dz dt d z d t. This case is analogous to the standard chain rule from Calculus I that we looked at above. In this case we are going to compute an ordinary derivative since z z really would be a function of t t only if we were to substitute in for x x and y y. Chain Rule. 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. For example, if a composite function f (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 ? . Chain Rule: The General Exponential Rule The exponential rule is a special case of the chain rule. It is useful when finding the derivative of e raised to the power of a function. The exponential rule states that this derivative is e to the power of the function times the derivative of the function. Derivatives of Exponential Functions.
However, if you look back they have all been functions similar to the following kinds of functions. These are all fairly simple functions in that wherever the variable appears it is by itself.
What about functions like the following,. Back in the section on the definition of the derivative we actually used the definition to compute this derivative. In that section we found that,. There are two forms of the chain rule. Here they are. Each of these forms have their uses, however we will work mostly with the first form in this class. In general, this is how we think of the chain rule.
We then differentiate the outside function leaving the inside function alone and multiply all of this by the derivative of the inside function. In its general form this is,. The square root is the last operation that we perform in the evaluation and this is also the outside function. The outside function will always be the last operation you would perform if you were going to evaluate the function. The derivative is then. In this case the outside function is the exponent of 50 and the inside function is all the stuff on the inside of the parenthesis.
In this case we need to be a little careful. Recall that the outside function is the last operation that we would perform in an evaluation. In this case if we were to evaluate this function the last operation would be the exponential. Therefore, the outside function is the exponential function and the inside function is its exponent. Remember, we leave the inside function alone when we differentiate the outside function. So, the derivative of the exponential function with the inside left alone is just the original function.
Here the outside function is the natural logarithm and the inside function is stuff on the inside of the logarithm. Again remember to leave the inside function alone when differentiating the outside function. There are two points to this problem. First, there are two terms and each will require a different application of the chain rule. Second, we need to be very careful in choosing the outside and inside function for each term. So, in the first term the outside function is the exponent of 4 and the inside function is the cosine.
There are a couple of general formulas that we can get for some special cases of the chain rule. The formulas in this example are really just special cases of the Chain Rule but may be useful to remember in order to quickly do some of these derivatives.
We will be assuming that you can see our choices based on the previous examples and the work that we have shown. However, in using the product rule and each derivative will require a chain rule application as well.
Now contrast this with the previous problem. In the previous problem we had a product that required us to use the chain rule in applying the product rule. In this case we did not actually do the derivative of the inside yet. We just left it in the derivative notation to make it clear that in order to do the derivative of the inside function we now have a product rule.
These tend to be a little messy. After factoring we were able to cancel some of the terms in the numerator against the denominator. So even though the initial chain rule was fairly messy the final answer is significantly simpler because of the factoring.
As with the second part above we did not initially differentiate the inside function in the first step to make it clear that it would be quotient rule from that point on.
There were several points in the last example. Just because we now have the chain rule does not mean that the product and quotient rule will no longer be needed. In addition, as the last example illustrated, the order in which they are done will vary as well. Some problems will be product or quotient rule problems that involve the chain rule. However, in practice they will often be in the same problem so you need to be prepared for these kinds of problems.
This is to allow us to notice that when we do differentiate the second term we will require the chain rule again. Notice as well that we will only need the chain rule on the exponential and not the first term. Be careful with the second application of the chain rule. As with the first example the second term of the inside function required the chain rule to differentiate it. Also note that again we need to be careful when multiplying by the derivative of the inside function when doing the chain rule on the second term.
In this example both of the terms in the inside function required a separate application of the chain rule. Sometimes these can get quite unpleasant and require many applications of the chain rule. Finally, before we move onto the next section there is one more issue that we need to address. In the Derivatives of Exponential and Logarithm Functions section we claimed that,. We now do. What we needed was the chain rule. This may seem kind of silly, but it is needed to compute the derivative.
Now, using this we can write the function as,. Now, differentiating the final version of this function is a hopefully fairly simple Chain Rule problem. Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width.
Example 2 Differentiate each of the following. Example 3 Differentiate each of the following. Example 4 Differentiate each of the following. Here is the chain rule portion of the problem.
Here is the rest of the work for this problem. Here is the work for this problem. Example 5 Differentiate each of the following.