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The Chain Rule. Chapter 3.4. The Chain Rule. The Chain Rule might perhaps be the most important of the differentiation rules It tells us how to differentiation a function within a function That is, it tells us how to differentiate a composition of functions
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The Chain Rule Chapter 3.4
The Chain Rule • The Chain Rule might perhaps be the most important of the differentiation rules • It tells us how to differentiation a function within a function • That is, it tells us how to differentiate a composition of functions • The next slide shows examples of functions that can be differentiated without the Chain Rule and those for which the Chain Rule can apply • Note that for those that use the Chain Rule, each can be thought of as a composition of functions
Theorem 3.11: The Chain Rule THEOREM: If is a differentiable function of u and is a differentiable function of x, then is a differentiable function of x and Or equivalently, PROOF We will define and use the alternative form of the derivative to show that, for
Theorem 3.11: The Chain Rule THEOREM: PROOF First, note that since by assumption, g is differentiable, then it is also continuous. So we know that and . We will also assume (the reason will be explained at the end) that there are no other values of x such that . Now by definition Here, we multiplied and divided by . It is this step that requires us to assume that for any values of x different from c.
Theorem 3.11: The Chain Rule THEOREM: PROOF Continuing
Theorem 3.11: The Chain Rule THEOREM: Although this proof seems reasonable, the problem is that we multiplied and divided by . This itself isn’t a problem if we are letting . But if there happen to be other values of x nearby such that (as we might find in a function that is not one-to-one), then we will have multiplied by the indefinite form . To avoid this, we could define two piecewise functions so that approach to c is continuous. Appendix A shows how this would work.
The Chain Rule • When applying the Chain Rule, it is important to identify the “inner” and the “outer” functions • If , then g is the “inner” function and f is the “outer” function • We may designate a function for that • The Chain Rule says that is the derivative of the “outer” function (as though were just a variable, say u) times the derivative of the “inner” function • That is,
The Chain Rule Using Notation • If we take and we take , then • In the last version, y is a function of u, so taking the derivative gives • However, u is a function of x, so its derivative is • We combine these to come up with the Chain Rule as follows
The Chain Rule Using Notation • Note that this looks like cross-canceling of du, but it really isn’t • The reason is that derivative is not a ratio, it is a limit • However, under the right circumstances it behaves as though it were a ratio • So, it’s ok to think about cross-canceling here, but do not assume without justification that these will behave just like ratios in any circumstance • For example, we should not jump to the conclusion that we can add
Example 3: Using the Chain Rule Find for . Our inner function here is while the outer function is . Find and and multiply them to get the answer: Don’t forget to back-substitute for u!
Example 3: Using the Chain Rule Find for . For this example we also could have applied the exponent and then taken the derivative (remember that this isn’t always possible and even when it is, it usually involves more work). So The result is the same.
The General Power Rule • In an earlier section of this chapter you saw that the Power Rule is • Using the Chain Rule, we can derive a more general Power Rule that applies whenever a function is raised to a power
Theorem 3.12: The General Power Rule THEOREM: If , where u is a differentiable function of x and n is a rational number, then Or equivalently The proof is a straightforward application of the Chain Rule
Example 4: Applying the General Power Rule Find the derivative of . Take . Then we have and the derivative is
Example 5: Differentiating Functions Involving Radicals Find all points on the graph of for which . It will be easier here to rewrite the function with a rational exponent: . Now take and . We have and . Applying the Chain Rule Note that if , then the numerator must be zero implying that . Hence, the point at which the slope of the tangent line is zero is (i.e., the value of ). Confirm this result using your graphing calculator.
Example 6: Differentiating Quotients With Constant Numerators Differentiate . Rewrite the function as . Now, and . So we have So the derivative is
Simplifying Derivatives • Finding derivatives of composite functions that also involve products or quotients requires keeping track of the various rules and where they must be applied in the function • It is usually best in these cases to break the derivative into its component parts and take derivatives separately as needed • It is best not to skimp on the use of notation
Example 7: Simplifying by Factoring Out the Least Powers Find the derivative of . First identify the operations in the function. Here we have a product and a radical which can be the “outer” function in a composite function. Begin by rewriting the radical with a rational exponent Apply the Product rule and use derivative notation for the composite part of the function Now let and and apply the Chain Rule Replace this in the original derivative and simplify
Example 7: Simplifying by Factoring Out the Least Powers Find the derivative of .
Example 8: Simplifying the Derivative of a Quotient Find the derivative of Rewrite the denominator with a rational exponent Apply the Quotient Rule Apply the Chain Rule with
Example 8: Simplifying the Derivative of a Quotient Find the derivative of Replace this in the original derivative and simplify
Example 9: Simplifying the Derivative of a Power Find the derivative of We can apply the General Power Rule Use the Quotient Rule Substitute this into the original derivative and simplify
Trigonometric Functions & the Chain Rule • The following show the application of the Chain Rule to trigonometric functions • Remember that these apply when the argument is itself a function • That is, in the table that follows, u will be a function of x
Example 10: Applying the Chain Rule to Transcendental Functions • , with . • , with . • , with .
Example 11: Parentheses & Trigonometric Functions • , with . • . Recognize that this is just a constant times , so the Chain Rule does not apply. We get • , with . • . Apply the General Power Rule:
Repeated Application of the Chain Rule • A function of the form requires two applications of the chain rule • To see why, let , , and • By the Chain Rule we get • In the next example, use u and v for the inner functions
Example 12: Repeated Application of the Chain Rule Find the derivative of First, rewrite the function as . Now take , , and . Differentiate each of these Remember that
Derivative of the Natural Logarithm Function • Up to this point, the derivatives of algebraic functions (like polynomial functions or radical functions) have been algebraic functions • Likewise, the derivatives of transcendental functions (like trigonometric functions and exponential functions) have been transcendental functions • The natural logarithm function is unusual in that it is a transcendental function, but its derivative is an algebraic function
Theorem 3.13: Derivative of the Natural Logarithm Function THEOREM: Let u be a differentiable function of x. Then PROOF Rewrite the function as . Now differentiate both sides of the equation with respect to x
Theorem 3.13: Derivative of the Natural Logarithm Function THEOREM: Note here that we have applied the Chain Rule to obtain the result on the left because y is a function of x. Another way to think of this is to set and we get Solve for
Example 13: Differentiation of the Logarithmic Functions • , with . • , with . • . Apply the Product Rule: • , with .
Logarithmic Properties as Aids to Differentiation • Logarithms were first developed by amateur mathematician John Napier during the 16th century as an aid to calculation, specifically multiplying, dividing, and raising to powers • Prior to the widespread availability of calculators, doing such calculations was tedious (as they still are) • Napier created tables of logarithmic values so that products and quotients could be turned into sums and differences using the properties of logarithms • Though such tables are now obsolete, we can use the logarithm properties to simplify differentiation, as the following examples show
Example 14: Logarithmic Properties as Aids to Differentiation Differentiate We could apply the Chain Rule twice here, but we can more easily make use of the logarithm properties by rewriting Now,
Example 15: Logarithmic Properties as Aids to Differentiation Differentiate Again we could apply the Chain Rule here, but using the logarithm properties greatly simplifies the task Differentiating
Derivative of the Natural Logarithm Involving Absolute Value • Since logarithms are undefined for , it is common to see the form • As the next theorem shows, we can differentiate such forms as though the absolute value signs were missing
Theorem 3.14: Derivative Involving Absolute Value THEOREM: If u is a differentiable function of x such that , then PROOF If , then and the result is the same as Theorem 3.13. If , then and we have
Exponential Function to Base a DEFINITION: If a is a positive real number () and x is any real number, then the exponential function to base ais denoted by as is defined by If , then is a constant function.
Logarithmic Function to Base a DEFINITION: If a is a positive real number () and x is any positive real number, then the logarithmic function to the base ais denoted by and is defined as Note that this can be thought of as the change of base formula applied to base a with input x.
Theorem 3.15: Derivatives for Bases Other Than e THEOREM: Let a be a positive real number other than 1 and let u be a differentiable function of x.
