Section 13.5 The Multivariable Chain Rule
Consider driving an off-road vehicle along a dirt road. As you drive, your elevation likely changes. What factors determine how quickly your elevation rises and falls? After some thought, generally one recognizes that one’s velocity (speed and direction) and the terrain influence your rise and fall.
One can represent the terrain as the surface defined by a multivariable function one can represent the path of the off-road vehicle, as seen from above, with a vector-valued function the velocity of the vehicle is thus
Consider Figure 13.5.1 in which a surface is drawn, along with a dashed curve in the -plane. Restricting to just the points on this circle gives the curve shown on the surface (i.e., “the path of the off-road vehicle.”) The derivative gives the instantaneous rate of change of with respect to If we consider an object traveling along this path, gives the rate at which the object rises/falls (i.e., “the rate of elevation change” of the vehicle.) Conceptually, the Multivariable Chain Rule combines terrain and velocity information properly to compute this rate of elevation change.
Abstractly, let be a function of and that is, for some function and let and each be functions of By choosing a -value, - and -values are determined, which in turn determine this defines as a function of The Multivariable Chain Rule gives a method of computing
Subsection 13.5.1 Multivariable Chain Rule, Part I
The Chain Rule of Section 2.5 states that
If we can express the Chain Rule as
recall that the derivative notation is deliberately chosen to reflect their fraction-like properties. A similar effect is seen in Theorem 13.5.3. In the second line of equations, one can think of the and as “sort of” canceling out, and likewise with and
Notice, too, the third line of equations in Theorem 13.5.3. The vector contains information about the surface (terrain); the vector can represent velocity. In the context measuring the rate of elevation change of the off-road vehicle, the Multivariable Chain Rule states it can be found through a product of terrain and velocity information.
We now practice applying the Multivariable Chain Rule.
Example 13.5.5. Using the Multivariable Chain Rule.
Solution.
This may look odd, as it seems that is a function of and Since and are functions of is really just a function of and we can replace with and with
The previous example can make us wonder: if we substituted for and at the end to show that is really just a function of why not substitute before differentiating, showing clearly that is a function of
This may now make one wonder “What’s the point? If we could already find the derivative, why learn another way of finding it?” In some cases, applying this rule makes deriving simpler, but this is hardly the power of the Chain Rule. Rather, in the case where and the Chain Rule is extremely powerful when we do not know what and/or are. It may be hard to believe, but often in “the real world” we know rate-of-change information (i.e., information about derivatives) without explicitly knowing the underlying functions. The Chain Rule allows us to combine several rates of change to find another rate of change. The Chain Rule also has theoretic use, giving us insight into the behavior of certain constructions (as we’ll see in the next section).
We demonstrate this in the next example.
Example 13.5.6. Applying the Multivariable Chain Rule.
An object travels along a path on a surface. The exact path and surface are not known, but at time it is known that :
We next apply the Chain Rule to solve a max/min problem.
Example 13.5.7. Applying the Multivariable Chain Rule.
Consider the surface a paraboloid, on which a particle moves with and coordinates given by and Find when and find where the particle reaches its maximum/minimum -values.
Solution.
We can use the First Derivative Test to find that on has reaches its absolute minimum at and it reaches its absolute maximum at and as shown in Figure 13.5.8.
We can extend the Chain Rule to include the situation where is a function of more than one variable, and each of these variables is also a function of more than one variable. The basic case of this is where and and are functions of two variables, say and
Theorem 13.5.9. Multivariable Chain Rule, Part II.
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Let
be a differentiable function of variables, where each of the is a differentiable function of the variables Then is a function of the and
Example 13.5.10. Using the Multivariable Chain Rule, Part II.
Solution.
Example 13.5.11. Using the Multivariable Chain Rule, Part II.
Solution.
Subsection 13.5.2 Implicit Differentiation
We studied finding when is given as an implicit function of in detail in Section 2.6. We find here that the Multivariable Chain Rule gives a simpler method of finding
Instead of using this method, consider The implicit function above describes the level curve Considering and as functions of the Multivariable Chain Rule states that
Note how our solution for in Equation (13.5.2) is just the partial derivative of with respect to divided by the partial derivative of with respect to all multiplied by
We state the above as a theorem.
Theorem 13.5.12. Implicit Differentiation.
Let be a differentiable function of and where defines as an implicit function of for some constant Then
Example 13.5.13. Implicit Differentiation.
Given the implicitly defined function find Note: this is the same problem as given in Example 2.6.8 of Section 2.6, where the solution took about a full page to find.
Solution.
Let the implicitly defined function above is equivalent to We find by applying Theorem 13.5.12. We find
so
which matches our solution from Example 2.6.8.
We can also do implicit differentiation for functions of three variables. In the same way that a level curve is used to implicitly define as a function of a level surface can be viewed as implicitly defining as a function of and
Suppose the equation where is a constant, defines the function Then we can use the chain rule to compute the derivatives of with respect to and where we set and Since is constant, we have
Solving for gives us
and similarly,
In Subsection 13.3.2 we saw that we can use partial derivatives to determine the equation of the tangent plane to a graph Using implicit differentiation, we can do the same for a level surface
Example 13.5.15. Implicit Differentiation with three variables.
Given that the equation
defines implicitly as a function of and compute and using implicit differentiation. Then, determine the equation of the tangent plane to the surface at the point
Solution.
There are two ways to proceed. One is to use implicit differentiation as before, but using partial derivatives. Whenever we differentiate a function of we multiply by the appropriate partial derivative of The other option is to use the formula derived above. We will use the first method for the derivative, and the second for
We first take the partial derivative of both sides of Equation (13.5.3) with respect to
Note that we treated as a constant, since the derivative is with respect to Next, we collect terms:
Lastly, we solve for
For the derivative, we will use the result given above. Setting we have Therefore,
The second method certainly seems simpler! The reader is invited to try each part with the other method, and compare answers.
Finally, we consider the problem of the tangent plane. First, we check that the point is indeed on the surface: as required. Next we note that is given to us from this point. So if implicitly defines the graph then we must have Next, we have
The equation of the tangent plane is therefore
In Section 13.3 we learned how partial derivatives give certain instantaneous rate of change information about a function In that section, we measured the rate of change of by holding one variable constant and letting the other vary (such as, holding constant and letting vary gives ). We can visualize this change by considering the surface defined by at a point and moving parallel to the -axis.
What if we want to move in a direction that is not parallel to a coordinate axis? Can we still measure instantaneous rates of change? Yes; we find out how in Section 13.6. In doing so, we’ll see how the Multivariable Chain Rule informs our understanding of these directional derivatives.
Exercises 13.5.3 Exercises
Terms and Concepts
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Put the blocks in order to form a correct chain rule statement.
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The Multivariable Chain Rule is only useful when all the related functions are known explicitly.
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Problems
Exercise Group.
Exercise Group.
In the following exercises, functions and are given. Find the values of where Note: these are the same surfaces/curves as found in Exercises 7–12.
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Exercise Group.
Exercise Group.
The given equation defines implicitly as a function of Find using Implicit Differentiation and Theorem 13.5.12.
Exercise Group.
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