# chain rule examples with solutions pdf

1. To avoid using the chain rule, recall the trigonometry identity , and first rewrite the problem as . This package reviews the chain rule which enables us to calculate the derivatives of functions of functions, such as sin(x3), and also of powers of functions, such as (5x2 −3x)17. (medium) Suppose the derivative of lnx exists. x��\Y��uN^����y�L�۪}1�-A�Al_�v S�D�u). Section 3-9 : Chain Rule. Tree diagrams are useful for deriving formulas for the chain rule for functions of more than one variable, where each independent variable also depends on other variables. The chain rule gives us that the derivative of h is . Chain Rule Examples (both methods) doc, 170 KB. %PDF-1.4 %���� If our function f(x) = (g◦h)(x), where g and h are simpler functions, then the Chain Rule may be stated as f′(x) = (g◦h) (x) = (g′◦h)(x)h′(x). dv dy dx dy = 18 8. dy dx + y 2. The Chain Rule (Implicit Function Rule) • If y is a function of v, and v is a function of x, then y is a function of x and dx dv. In this unit we will refer to it as the chain rule. h�bf��������A��b�,;>���1Y���������Z�b��k���V���Y��4bk�t�n W�h���}b�D���I5����mM꺫�g-��w�Z�l�5��G�t� ��t�c�:��bY��0�10H+$8�e�����˦0]��#��%llRG�.�,��1��/]�K�ŝ�X7@�&��X�����  %�bl endstream endobj 58 0 obj <> endobj 59 0 obj <> endobj 60 0 obj <>stream 4 Examples 4.1 Example 1 Solve the differential equation 3x2y00+xy0 8y=0. SOLUTION 20 : Assume that , where f is a differentiable function. h�bbdb^$��7 H0���D�S�|@�#���j@��Ě"� �� �H���@�s!H��P�$D��W0��] �ʆ�f��7w������ٴ"L��,���Jڜ �X��0�mm�%�h�tc� m�p}��J�b�f�4Q��XXЛ�p0��迒1�A��� eܟN�{P������1��\XL�O5M�ܑw��q��)D0����a�\�R(y�2s�B� ���|0�e����'��V�?��꟒���d� a躆�i�2�6�J�=���2�iW;�Mf��B=�}T�G�Y�M�. Chain Rule Worksheets with Answers admin October 1, 2019 Some of the Worksheets below are Chain Rule Worksheets with Answers, usage of the chain rule to obtain the derivatives of functions, several interesting chain rule exercises with step by step solutions and quizzes with answers, … Using the linear properties of the derivative, the chain rule and the double angle formula , we obtain: {y’\left( x \right) }={ {\left( {\cos 2x – 2\sin x} \right)^\prime } } Click HERE to return to the list of problems. Section 1: Partial Diﬀerentiation (Introduction) 5 The symbol ∂ is used whenever a function with more than one variable is being diﬀerentiated but the techniques of partial … Solution: Step 1 d dx x2 + y2 d dx 25 d dx x2 + d dx y2 = 0 Use: d dx y2 = d dx f(x) 2 = 2f(x) f0(x) = 2y y0 2x + 2y y0= 0 Step 2 That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function f ∘ g in terms of the derivatives of f and g. Definition •If g is differentiable at x and f is differentiable at g(x), … Created: Dec 4, 2011. by the Chain Rule, dy/dx = dy/dt × dt/dx so dy/dx = 3t² × 2x = 3(1 + x²)² × 2x = 6x(1 + x²)². Hyperbolic Functions And Their Derivatives. Show all files. Revision of the chain rule We revise the chain rule by means of an example. Solution: This problem requires the chain rule. 2 1 0 1 2 y 2 10 1 2 x Figure 21: The hyperbola y − x2 = 1. •Prove the chain rule •Learn how to use it •Do example problems . Scroll down the page for more examples and solutions. After having gone through the stuff given above, we hope that the students would have understood, "Chain Rule Examples With Solutions" Apart from the stuff given in "Chain Rule Examples With Solutions", if you need any other stuff in math, please use our google custom search here. %PDF-1.4 View Notes - Introduction to Chain Rule Solutions.pdf from MAT 122 at Phoenix College. Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule. The chain rule provides a method for replacing a complicated integral by a simpler integral. Definition •In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. Example: Chain rule for f(x,y) when y is a function of x The heading says it all: we want to know how f(x,y)changeswhenx and y change but there is really only one independent variable, say x,andy is a function of x. SOLUTION 6 : Differentiate . Learn its definition, formulas, product rule, chain rule and examples at BYJU'S. d dx (e3x2)= deu dx where u =3x2 = deu du × du dx by the chain rule = eu × du dx = e3x2 × d dx (3x2) =6xe3x2. ��#�� Solution: This problem requires the chain rule. The problem that many students have trouble with is trying to figure out which parts of the function are within other functions (i.e., in the above example, which part if g(x) and which part is h(x). Example Diﬀerentiate ln(2x3 +5x2 −3). Example Find d dx (e x3+2). Scroll down the page for more examples and solutions. 2 1 0 1 2 y 2 10 1 2 x Figure 21: The hyperbola y − x2 = 1. Step 1. The Chain Rule for Powers 4. This rule is obtained from the chain rule by choosing u … Example Find d dx (e x3+2). We must identify the functions g and h which we compose to get log(1 x2). This section shows how to differentiate the function y = 3x + 1 2 using the chain rule. If f(x) = g(h(x)) then f0(x) = g0(h(x))h0(x). dv dy dx dy = 17 Examples i) y = (ax2 + bx)½ let v = (ax2 + bx) , so y = v½ ()ax bx . 2.Write y0= dy dx and solve for y 0. Then (This is an acceptable answer. Calculate (a) D(y 3), (b) d dx (x 3 y 2), and (c) (sin(y) )' Solution: (a) We need the Power Rule for Functions since y is a function of x: D(y 3) = 3 y 2. x + dx dy dx dv. We illustrate with an example: 35.1.1 Example Find Z cos(x+ 1)dx: Solution We know a rule that comes close to working here, namely, R cosxdx= sinx+C, The Rules of Partial Diﬀerentiation Since partial diﬀerentiation is essentially the same as ordinary diﬀer-entiation, the product, quotient and chain rules may be applied. For example, all have just x as the argument. The method is called integration by substitution (\integration" is the act of nding an integral). As we apply the chain rule, we will always focus on figuring out what the “outside” and “inside” functions are first. Then . A good way to detect the chain rule is to read the problem aloud. The symbol dy dx is an abbreviation for ”the change in y (dy) FROM a change in x (dx)”; or the ”rise over the run”. Show Solution. Let so that (Don't forget to use the chain rule when differentiating .) The inner function is the one inside the parentheses: x 2 -3. We must identify the functions g and h which we compose to get log(1 x2). 1. Now apply the product rule. If you have any feedback about our math content, please mail us : v4formath@gmail.com. Use the solutions intelligently. rule d y d x = d y d u d u d x ecomes Rule) d d x f ( g ( x = f 0 ( g ( x )) g 0 ( x ) \outer" function times of function. Then . Then if such a number λ exists we deﬁne f′(a) = λ. From there, it is just about going along with the formula. Multi-variable Taylor Expansions 7 1. Basic Results Diﬀerentiation is a very powerful mathematical tool. Chain rule examples: Exponential Functions. Example: a) Find dy dx by implicit di erentiation given that x2 + y2 = 25. The chain rule 2 4. In Leibniz notation, if y = f (u) and u = g (x) are both differentiable functions, then Note: In the Chain Rule, we work from the outside to the inside. This video gives the definitions of the hyperbolic functions, a rough graph of three of the hyperbolic functions: y = sinh x, y = … For example: 1 y = x2 2 y =3 √ x =3x1/2 3 y = ax+bx2 +c (2) Each equation is illustrated in Figure 1. y y y x x Y = x2 Y = x1/2 Y = ax2 + bx Figure 1: 1.2 The Derivative Given the general function y = f(x) the derivative of y is denoted as dy dx = f0(x)(=y0) 1. The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables. Since the functions were linear, this example was trivial. We always appreciate your feedback. , or . Section 2: The Rules of Partial Diﬀerentiation 6 2. When the argument of a function is anything other than a plain old x, such as y = sin (x 2) or ln10 x (as opposed to ln x), you’ve … Although the chain rule is no more com-plicated than the rest, it’s easier to misunderstand it, and it takes care to determine whether the chain rule or the product rule is needed. if x f t= ( ) and y g t= ( ), then by Chain Rule dy dx = dy dx dt dt, if 0 dx dt ≠ Chapter 6 APPLICATION OF DERIVATIVES APPLICATION OF DERIVATIVES 195 Thus, the rate of change of y with respect to x can be calculated using the rate of change of y and that of x both with respect to t. Let us consider some examples. Solution: d d x sin( x 2 os( x 2) d d x x 2 =2 x cos( x 2). Solution. √ √Let √ inside outside functionofafunction. To write the indicial equation, use the TI-Nspire CAS constraint operator to substitute the values of the constants in the symbolic … Example 1: Assume that y is a function of x . The outer layer of this function is the third power'' and the inner layer is f(x) . dx dg dx While implicitly diﬀerentiating an expression like x + y2 we use the chain rule as follows: d (y 2 ) = d(y2) dy = 2yy . Chain rule. Find the derivative of y = 6e7x+22 Answer: y0= 42e7x+22 In fact we have already found the derivative of g(x) = sin(x2) in Example 1, so we can reuse that result here. How to use the Chain Rule. Solution: Using the above table and the Chain Rule. It’s no coincidence that this is exactly the integral we computed in (8.1.1), we have simply renamed the variable u to make the calculations less confusing. H��TMo�0��W�h'��r�zȒl�8X����+NҸk�!"�O�|��Q�����&�ʨ���C�%�BR��Q��z;�^_ڨ! For instance, if f and g are functions, then the chain rule expresses the derivative of their composition. The same is true of our current expression: Z x2 −2 √ u du dx dx = Z x2 −2 √ udu. There is also another notation which can be easier to work with when using the Chain Rule. Hyperbolic Functions - The Basics. Work through some of the examples in your textbook, and compare your solution to the detailed solution o ered by the textbook. Click HERE to return to the list of problems. The following chain rule examples show you how to differentiate (find the derivative of) many functions that have an “ inner function ” and an “ outer function.” For an example, take the function y = √ (x 2 – 3). Info. doc, 90 KB. It is often useful to create a visual representation of Equation for the chain rule. In this presentation, both the chain rule and implicit differentiation will /� �؈L@'ͱ݌�z���X�0�d\�R��9����y~c Target: On completion of this worksheet you should be able to use the chain rule to differentiate functions of a function. Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule. Both use the rules for derivatives by applying them in slightly different ways to differentiate the complex equations without much hassle. y = x3 + 2 is a function of x y = (x3 + 2)2 is a function (the square) of the function (x3 + 2) of x. The Total Derivative Recall, from calculus I, that if f : R → R is a function then f ′(a) = lim h→0 f(a+h) −f(a) h. We can rewrite this as lim h→0 f(a+h)− f(a)− f′(a)h h = 0. �x$�V �L�@na%�'�3� 0 �0S endstream endobj startxref 0 %%EOF 151 0 obj <>stream About this resource. Thus p = kT 1 V = kTV−1, ∴ ∂p ∂V = −kTV−2 = − kT V2. Click HERE to return to the list of problems. SOLUTION 8 : Integrate . For example, they can help you get started on an exercise, or they can allow you to check whether your intermediate results are correct Try to make less use of the full solutions as you work your way ... using the chain rule, and dv dx = (x+1) A transposition is a permutation that exchanges two cards. Most of the basic derivative rules have a plain old x as the argument (or input variable) of the function. Show Solution For this problem the outside function is (hopefully) clearly the exponent of 4 on the parenthesis while the inside function is the polynomial that is being raised to the power. The Chain Rule for Powers The chain rule for powers tells us how to diﬀerentiate a function raised to a power. du dx Chain-Log Rule Ex3a. NCERT Books. NCERT Books for Class 5; NCERT Books Class 6; NCERT Books for Class 7; NCERT Books for Class 8; NCERT Books for Class 9; NCERT Books for Class 10; NCERT Books for Class 11; NCERT … <> There is a separate unit which covers this particular rule thoroughly, although we will revise it brieﬂy here. Written this way we could then say that f is diﬀerentiable at a if there is a number λ ∈ R such that lim h→0 f(a+h)− f(a)−λh h = 0. We ﬁrst explain what is meant by this term and then learn about the Chain Rule which is the technique used to perform the diﬀerentiation. We are nding the derivative of the logarithm of 1 x2; the of almost always means a chain rule. stream Introduction In this unit we learn how to diﬀerentiate a ‘function of a function’. A function of a … In applying the Chain Rule, think of the opposite function f °g as having an inside and an outside part: General Power Rule a special case of the Chain Rule. For this problem the outside function is (hopefully) clearly the exponent of 4 on the parenthesis while the inside function is the polynomial that is being raised to the power. Differentiation Using the Chain Rule. 3x 2 = 2x 3 y. dy … The rule is given without any proof. u and the chain rule gives df dx = df du du dv dv dx = cosv 3u2=3 1 3x2=3 = cos 3 p x 9(xsin 3 p x)2=3: 11. (a) z … After having gone through the stuff given above, we hope that the students would have understood, "Chain Rule Examples With Solutions"Apart from the stuff given in "Chain Rule Examples With Solutions", if you need any other stuff in math, please use our google custom search here. Example 1 Find the rate of change of the area of a circle per second with respect to its … If f(x) = g(h(x)) then f0(x) = g0(h(x))h0(x). Title: Calculus: Differentiation using the chain rule. Solution This is an application of the chain rule together with our knowledge of the derivative of ex. %�쏢 2. Solution: The derivatives of f and g aref′(x)=6g′(x)=−2.According to the chain rule, h′(x)=f′(g(x))g′(x)=f′(−2x+5)(−2)=6(−2)=−12. For functions f and g d dx [f(g(x))] = f0(g(x)) g0(x): In the composition f(g(x)), we call f the outside function and g the inside function. This is called a tree diagram for the chain rule for functions of one variable and it provides a way to remember the formula (Figure $$\PageIndex{1}$$). Differentiation Using the Chain Rule. You must use the Chain rule to find the derivative of any function that is comprised of one function inside of another function. Usually what follows Find the derivative of $$f(x) = (3x + 1)^5$$. The more times you apply the chain rule to different problems, the easier it becomes to recognize how to apply the rule. If , where u is a differentiable function of x and n is a rational number, then Examples: Find the derivative of … The derivative is then, \[f'\left( x \right) = 4{\left( {6{x^2} + 7x} \right)^3}\left( … A simple technique for diﬀerentiating directly 5 www.mathcentre.ac.uk 1 c mathcentre 2009. Many answers: Ex y = (((2x + 1)5 + 2) 6 + 3) 7 dy dx = 7(((2x + 1)5 + 2) 6 + 3) 6 ⋅ 6((2x + 1)5 + 2) 5 ⋅ 5(2x + 1)4 ⋅ 2-2-Create your own worksheets like this … Solution: Using the table above and the Chain Rule. If , where u is a differentiable function of x and n is a rational number, then Examples: Find the derivative of each function given below. Use the chain rule to calculate h′(x), where h(x)=f(g(x)). The chain rule gives us that the derivative of h is . Solution. In applying the Chain Rule, think of the opposite function f °g as having an inside and an outside part: General Power Rule a special case of the Chain Rule. Moveover, in this case, if we calculate h(x),h(x)=f(g(x))=f(−2x+5)=6(−2x+5)+3=−12x+30+3=−12… The Inverse Function Rule • Examples If x = f(y) then dy dx dx dy 1 = i) … 13) Give a function that requires three applications of the chain rule to differentiate. BOOK FREE CLASS; COMPETITIVE EXAMS. Usually what follows Examples using the chain rule. The following figure gives the Chain Rule that is used to find the derivative of composite functions. Solution Again, we use our knowledge of the derivative of ex together with the chain rule. d dx (ex3+2x)= deu dx (where u = x3 +2x) = eu × du dx (by the chain rule) = ex3+2x × d dx (x3 +2x) =(3x2 +2)×ex3+2x. Section 3: The Chain Rule for Powers 8 3. Use u-substitution. To avoid using the chain rule, first rewrite the problem as . Differentiate algebraic and trigonometric equations, rate of change, stationary points, nature, curve sketching, and equation of tangent in Higher Maths. 5 0 obj Does your textbook come with a review section for each chapter or grouping of chapters? Substitute into the original problem, replacing all forms of , getting . Chain Rule Examples (both methods) doc, 170 KB. Some examples involving trigonometric functions 4 5. Let f(x)=6x+3 and g(x)=−2x+5. (b) We need to use the product rule and the Chain Rule: d dx (x 3 y 2) = x 3. d dx (y 2) + y 2. d dx (x 3) = x 3 2y. dx dy dx Why can we treat y as a function of x in this way? 1.3 The Five Rules 1.3.1 The … Example 2. by the Chain Rule, dy/dx = dy/dt × dt/dx so dy/dx = 3t² × 2x = 3(1 + x²)² × 2x = 6x(1 + x²)². The following examples demonstrate how to solve these equations with TI-Nspire CAS when x >0. Example Suppose we wish to diﬀerentiate y = (5+2x)10 in order to calculate dy dx. Final Quiz Solutions to Exercises Solutions to Quizzes. This diagram can be expanded for functions of more than one variable, as we shall see very shortly. In examples such as the above one, with practise it should be possible for you to be able to simply write down the answer without having to let t = 1 + x² etc. Example 3 Find ∂z ∂x for each of the following functions. For the matrices that are stochastic matrices, draw the associated Markov Chain and obtain the steady state probabilities (if they exist, if If 30 men can build a wall 56 meters long in 5 days, what length of a similar wall can be built by 40 … If and , determine an equation of the line tangent to the graph of h at x=0 . If and , determine an equation of the line tangent to the graph of h at x=0 . D(y ) = 3 y 2. y '. Updated: Mar 23, 2017. doc, 23 KB. Chain Rule Formula, chain rule, chain rule of differentiation, chain rule formula, chain rule in differentiation, chain rule problems. dx dg dx While implicitly diﬀerentiating an expression like x + y2 we use the chain rule as follows: d (y 2 ) = d(y2) dy = 2yy . The Chain Rule 4 3. Differentiating using the chain rule usually involves a little intuition. Even though we had to evaluate f′ at g(x)=−2x+5, that didn't make a difference since f′=6 not matter what its input is. To avoid using the chain rule, first rewrite the problem as . A good way to detect the chain rule is to read the problem aloud. The difficulty in using the chain rule: Implementing the chain rule is usually not difficult. ()ax b dx dy = + + − 2 2 1 2 1 2 ii) y = (4x3 + 3x – 7 )4 let v = (4x3 + 3x – 7 ), so y = v4 4()(4 3 7 12 2 3) = x3 + x − 3 . To differentiate this we write u = (x3 + 2), so that y = u2 Example. The rules of di erentiation are straightforward, but knowing when to use them and in what order takes practice. Study the examples in your lecture notes in detail. Let Then 2. The best way to memorize this (along with the other rules) is just by practicing until you can do it without thinking about it. The outer layer of this function is the third power'' and the inner layer is f(x) . Derivative Rules - Constant Rule, Constant Multiple Rule, Power Rule, Sum Rule, Difference Rule, Product Rule, Quotient Rule, Chain Rule, Exponential Functions, Logarithmic Functions, Trigonometric Functions, Inverse Trigonometric Functions, Hyperbolic Functions and Inverse Hyperbolic Functions, with video lessons, examples and step-by-step solutions. To avoid using the chain rule, recall the trigonometry identity , and first rewrite the problem as . Make use of it. It states: if y = (f(x))n, then dy dx = nf0(x)(f(x))n−1 where f0(x) is the derivative of f(x) with respect to x. We are nding the derivative of the logarithm of 1 x2; the of almost always means a chain rule. 'ɗ�m����sq'�"d����a�n�����R��� S>��X����j��e�\�i'�9��hl�֊�˟o��[1dv�{� g�?�h��#H�����G��~�1�yӅOXx�. Find it using the chain rule. Take d dx of both sides of the equation. This 105. is captured by the third of the four branch diagrams on the previous page. SOLUTION 9 : Integrate . dx dy dx Why can we treat y as a function of x in this way? [��IX��I� ˲ H|�d��[z����p+G�K��d�:���j:%��U>����nm���H:�D�}�i��d86c��b���l��J��jO[�y�Р�"?L��{.����C�f It’s also one of the most used. SOLUTION 20 : Assume that , where f is a differentiable function. Example: Find d d x sin( x 2). Then differentiate the function. 2. The outer function is √ (x). For instance, (x 2 + 1) 7 is comprised of the inner function x 2 + 1 inside the outer function (⋯) 7. Now we’re almost there: since u = 1−x2, x2 = 1− u and the integral is Z − 1 2 (1−u) √ udu. !w�@�����Bab��JIDу>�Y"�Ĉ2FP;=�E���Y��Ɯ��M��3f�+o��DLp�[BEGG���#�=a���G�f�l��ODK�����oDǟp&�)�8>ø�%�:�>����e �i":���&�_�J�\�|�h���xH�=���e"}�,*�����N8l��y���:ZY�S�b{�齖|�3[�Zk���U�6H��h��%�T68N���o��� 2.5 The Chain Rule Brian E. Veitch 2.5 The Chain Rule This is our last di erentiation rule for this course. Ask yourself, why they were o ered by the instructor. Notice that there are exactly N 2 transpositions. Now apply the product rule twice. For example, in Leibniz notation the chain rule is dy dx = dy dt dt dx. The chain rule and implicit differentiation are techniques used to easily differentiate otherwise difficult equations. … Now apply the product rule. The chain rule is probably the trickiest among the advanced derivative rules, but it’s really not that bad if you focus clearly on what’s going on. Section 1: Basic Results 3 1. BNAT; Classes. (b) For this part, T is treated as a constant. 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Just as before: … {�F?р��F���㸝.�7�FS������V��zΑm���%a=;^�K��v_6���ft�8CR�,�vy>5d륜��UG�/��A�FR0ם�'�,u#K �B7~���1&��|��J�ꉤZ���GV�q��T��{����70"cU�������1j�V_U('u�k��ZT. differentiate and to use the Chain Rule or the Power Rule for Functions. Ok, so what’s the chain rule? Write the solutions by plugging the roots in the solution form. General Procedure 1. In examples such as the above one, with practise it should be possible for you to be able to simply write down the answer without having to let t = 1 + x² etc. The random transposition Markov chain on the permutation group SN (the set of all permutations of N cards) is a Markov chain whose transition probabilities are p(x,˙x)=1= N 2 for all transpositions ˙; p(x,y)=0 otherwise. Example: Find the derivative of . As another example, e sin x is comprised of the inner function sin Then (This is an acceptable answer. 6 f ' x = 56 2x - 7 1. f x 4 2x 7 2. f x 39x 4 3. f x 3x 2 7 x 2 4. f x 4 x 5x 7 f ' x = 4 5 5 -108 9x - Now apply the product rule twice. For problems 1 – 27 differentiate the given function. Class 1 - 3; Class 4 - 5; Class 6 - 10; Class 11 - 12; CBSE. In other words, the slope. Implicit Diﬀerentiation and the Chain Rule The chain rule tells us that: d df dg (f g) = . Example: Differentiate . SOLUTION 6 : Differentiate . Solution Again, we use our knowledge of the derivative of ex together with the chain rule. Solution (a) This part of the example proceeds as follows: p = kT V, ∴ ∂p ∂T = k V, where V is treated as a constant for this calculation. The Chain Rule is a formula for computing the derivative of the composition of two or more functions. Implicit Diﬀerentiation and the Chain Rule The chain rule tells us that: d df dg (f g) = . This might … Df dg ( f g ) = λ rule when differentiating. 4 - 5 ; 4! Us that the derivative of their composition Figure gives the chain rule to differentiate functions a. A very powerful mathematical tool we treat y as a function of x in this way the difficulty in the! The of almost always means a chain rule to find the derivative of ex together with the.. Differentiation are techniques used to find the derivative of composite functions a little.... Thoroughly, although we will refer to it as the argument the following Figure the...: Z x2 −2 √ u du dx dx = Z x2 −2 u! Veitch 2.5 the chain rule problems > 0 the act of nding an integral ) to work with using. This example was trivial this 105. is captured by the third power '' and the inner layer f! 3X2Y00+Xy0 8y=0 for diﬀerentiating directly 5 www.mathcentre.ac.uk 1 c mathcentre 2009 forget to use the chain rule means... On completion of this function is the one inside the parentheses: x 2 -3 unit. 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