Chain Rule
The chain rule is a fundamental concept in calculus that allows for the differentiation of composite functions. It states that the derivative of a composite function is the product of the derivatives of the outer and inner functions. This rule is essential for finding the derivative of complex functions and is widely used in various fields of mathematics and science.
5 Key excerpts on "Chain Rule"
- Stu Schwartz(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Rather than trying to identify the form of a function (such as product or quotient), and then applying the derivative rule, the Chain Rule is a technique that oversees all the rules of differentiation. The Chain Rule is always in effect! The Chain Rule can be expressed several ways and can be confusing to students. Like trying to explain how to ride a bike, sometimes it is easier just to try it. if y = f (x) and u = g (x), EXAMPLE 13: If, find f′ (x) and f″ (x). SOLUTION: f ′(x) = (4 x + 1) 1/2 The differentiation technique requires you to treat the (4 x + 1) 1/2 as an entity and apply the power rule to it. EXAMPLE 14: If, find the slope of the line normal to the graph of y at x = 3. SOLUTION: The expression in brackets is the derivative of using the quotient rule. Slope of normal line at x = 3 is. EXAMPLE 15: The table below gives values of the functions f and g and their derivatives at selected values. of x with a being a constant. If the slope of the tangent line to f (g (x)) at x = 1 is 5, find the value of a. SOLUTION: DID YOU KNOW? There are many different notations for derivatives. The earliest,, was introduced by Gottfried Wilhelm Leibniz (1646−1716). Joseph-Louis Lagrange (1736− 1813) introduced the prime notation such as f′ and f″. However there are several more obscure notations that still survive. Leonhard Euler (1707-1873) used the notation D x y as the first derivative and D 2 x y as the second derivative. Sir Isaac Newton (1642−1727) used dot notation, representing the first two derivatives as and. Tangent Line Approximations Overview : Now that we can calculate the slope of the tangent line by computing the derivative of f and evaluating it at some value of x, we take the next step and actually find the equation of the tangent line. Recall that if we have a graph of y = f (x) passing through a point (x 1, y 1), the equation of the tangent line using the point - slope formula is: y − y 1 = m (x − x 1)...
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...It is not uncommon to need to apply both the product rule and the Chain Rule in the same problem, as seen in the following example. EXAMPLE 4.16 Find if y = ln(x 2 + 1) · tan(4 x 3). SOLUTION The product rule is used first, and the derivatives in the product rule use the Chain Rule. Sometimes the Chain Rule may need to be applied multiple times in a row within a problem. This occurs when the composition of more than two functions can be identified. The following example shows both ways of handling a situation such as this. It is done by decomposition into individual functions and by multiple use of the outside-inside principle. EXAMPLE 4.17 If find SOLUTION Decompose the function into a cubic, a trigonometric, and a radical function. Let y = u 3, u = sin(v), 1 and Substitution back to the original variable is almost always required when using the Chain Rule. It is sometimes easier to rewrite powers of trigonometric functions prior to using the outside-inside principle. Think of as The outermost function is the cubic function. The next function “in” is the sine function. And the innermost function is the radical. With the introduction of the Chain Rule into the course material, all the previously given derivative formulas take on a somewhat new look, each with the Chain Rule multiplier as a factor. Table 4.2 summarizes those formulas, each of which must be memorized. In all cases, u and v are differentiable functions of an unnamed independent variable; du is the Chain Rule derivative of that function; and k, n, and a are constants. Table 4.2 Function Derivative y = u n (n is any real number) dy = n · u (n– 1) · du y = kn (k is a constant) dy = 0 y = u · v dy = u · dv + v · du y = f (g (x)) y = sin(u) dy = cos(u) · du y = cos(u) dy = –. sin(u) · du y = tan(u) dy = sec 2 (u) · du y = cot(u) dy = – csc 2 (u) · du y = sec(u) dy = sec(u) · tan(u) du y = csc(u) dy = – csc(u) · cot(u) du y = e µ dy = e µ · du y = a u dy = a u · ln(a) · du y = ln(u) y = log a u HIGHER-ORDER...
eBook - ePub- Arsen Melkumian(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...and = a. Thus E XAMPLE 4.9 Using the Chain Rule find when y = and u = 3 x 2 + 2 x + 17. Solution : Note that = u − = 6 x + 2. So Recall that derivative represents rate of change. In a situation when we have quantity y depending on another quantity u, which in turn depends on another quantity x, we expect the total rate of change, y with respect to x, to be the product of the rates. That is the gist of the Chain Rule. PROBLEMS FOR SECTION 4.3 1. Find the first derivative of the following functions. (a) y(x) = 5 x 4 − 4 x 3 + 3 x 2 − 21 x + 15 (b) g(x) = 5 e x + ln x − 2 x 0.5 (c) (d) k(x) = (x 3 + 1)(x 2 + 1)(x + 1)a (e) w(x) = ax 4 + b(x + 1)(cx 2 + 2) (f) (g) f (x) = (e x + 1)(ln x + 2)(x 4 + 3) (h) (i) t (x) = (x 2 +. 3)(x 3 − 4)(x 4 + 5)(x 5 − 6)(x 6 + 7) 2. Find the first and second derivatives of the following functions. (a) y(x) = 14 x 3 + 3 x 2 − 21 x + 23 (b) (c) h(x) = (ax 2 − b)(cx 5 + dx) (d) (e) t (x) = x −5 (x 4 + 1) (f) (g) (h) (i) (j) w(x) = 5 x −5 + x −4 (k) 3. Use the Chain Rule to find for the following functions. (a) (b) y =. (x 5 + x 2) 99 (c) (d) (e) y = (25 x + x 25) 3 (f) (g) (h) y = 5 + (x 5 + x 4 + x 3 + x 2 + x + 1) 6 (i) (j) y = (2 x 4 + 5) 88 4. Use the Chain Rule to find for the following functions. (a) (b) y = 7[21 + (3 x 2 + 1) 40 ] 85 (c) (d) y =[72 x + (x 16 − 2 x 3) 6 ] (e) y = 1 +[1 +. (x 2 + 1) 5 ] 5 5. Let W = h(k) and k = f (t). Find the formula for 6. Let F = g(v) and v = f (t). Find the formula for 7. Suppose that the total cost function is TC = 400 − 3(525 − 3 Q), where Q ≤ 70. Use the Chain Rule to find 4.4 Total and marginal functions A total cost function TC(Q) assigns to each level of output Q the total cost of producing that level of output. Naturally, total cost functions are strictly increasing functions and The derivative MC(Q) of the total cost function is known as the marginal cost function...
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...This is an example of a pair of nested functions, a fact that becomes a little more apparent by inventing a new function of x, which we will call u (x). If we set u (x) equal to the expression inside the square root, so that u = A + B x, then we can rewrite the equation for y in terms of u : y = u 1 / 2 − C. By applying the rules encountered in Sections 5.4 and 5.5, we can work out the derivative of u with respect to x, and the derivative of y with respect to the new variable u : d u d x = B and d y d u = 1 2 u − 1 / 2. However, to get an expression for the derivative of y with respect to the original variable x, we need help, which is where the Chain Rule comes in. (Box 7.2). If y is a function of u (that is, y = f (u)), and u is itself a function of the variable x (that is, u = g (x), so that y = f (u) = f (g (x))), then the Chain Rule states that: d y d x = d y d u × d u d x. (EQ7.6) Returning to the acetic acid problem, we can make use of the Chain Rule to work out the rate of change of the concentration of hydrogen ions at any given concentration of. acid: d y d x = d y d u × d u d x = 1 2 u − 1 / 2 × B. Substituting u = A + Bx back into this equation will give an expression written in terms of y and x only: d y d x = 1 2 (A + B x) − 1 / 2 × B = B 2 A + B x. Box 7.2 The Chain Rule If y is a function of u and u is a function of x, then a small increment δ x in the. variable x will cause corresponding increments δ u and δ y in u and y, respectively. Using the same kind of approximation introduced in Section 5.11, δ y can be expressed in terms of δ u by using the derivative of y with respect to u. Similarly, δ u can also be expressed in terms of δ x : δ y ≈ d y d u δ u and δ u ≈ d u d x δ x. So, substituting for δ u, we get δ y ≈ d y d u × d u d x δ x. Dividing both sides of this equation by δ x therefore gives: δ y δ x ≈ d y d u d u d x. In the limit where δ x tends toward zero, the relation becomes...
- J. Rosebush, Stu Schwartz(Authors)
- 2016(Publication Date)
- Research & Education Association(Publisher)
...So, if equals one of these indeterminate forms, then take. Note that you are not using the quotient rule here, you are simply taking the derivative of the numerator and denominator separately. If the limit still has an indeterminate form then repeat the process as necessary. This also applies to cases in which x → ±∞. 1. For example,. Using L’Hôspital’s rule,. G. Derivative Rules 1. When taking the derivative of a function you might have to use more than one of the above rules. 2. There are some functions whose derivatives occur very often on the exam and it would save you time if memorized. These are the derivatives of and more generally, ; and and more generally,. Note that the Chain Rule was used in both general cases. H. Derivatives of trigonometric functions 1. The derivatives of the cofunctions are negative. 2. In taking the derivative of most trigonometric functions you will need to use the Chain Rule since most will be compositions—sometimes of more than two functions. Here is an example of the derivative of a function of the form y = f (g (h)): y = sin(tan(x 2)) → y ′ = cos(tan(x 2))sec 2 (x 2)(2 x). I. Derivatives of inverse trigonometric functions 1. Note that the derivatives of the cofunctions are the negatives of the derivatives of the functions. 2. In most cases, the Chain Rule is used. For example,. J. Implicit Differentiation—this means finding y ′ when the equation given is not explicitly defined in terms of y (that is, it is not of the form y = f (x)). In this case you must remember to always use the Chain Rule when taking the derivative of an expression involving y. That is all! Example 1 : Find y ′ if x 2 + y 2 = 3. Taking derivatives on both sides, 2 x + 2 yy ′ = 0 →. Example 2: Find y ′ if x 2 y 2 – 3 ln y = x + 7. Taking derivatives on both sides,...
