chain rule examples with solutions pdf

The rule is given without any proof. Differentiating using the chain rule usually involves a little intuition. 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. To write the indicial equation, use the TI-Nspire CAS constraint operator to substitute the values of the constants in the symbolic … �`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�. Example Find d dx (e x3+2). Show all files. 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. Chain Rule Examples (both methods) doc, 170 KB. SOLUTION 9 : Integrate . To avoid using the chain rule, recall the trigonometry identity , and first rewrite the problem as . Example: Find the derivative of . The chain rule 2 4. Show Solution. It is convenient … Now apply the product rule. Step 1. Ask yourself, why they were o ered by the instructor. 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”. Study the examples in your lecture notes in detail. h�bbd``b`^$��7 H0��D�S�|@�#��j@��Ě"� �� H��@�s!H��P�$D��W0��] Let Then 2. 3x 2 = 2x 3 y. dy … SOLUTION 6 : Differentiate . 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). We must identify the functions g and h which we compose to get log(1 x2). 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. 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. doc, 90 KB. In this unit we will refer to it as the chain rule. For this equation, a = 3;b = 1, and c = 8. We are nding the derivative of the logarithm of 1 x2; the of almost always means a chain rule. differentiate and to use the Chain Rule or the Power Rule for Functions. There is a separate unit which covers this particular rule thoroughly, although we will revise it brieﬂy here. For example, all have just x as the argument. , or . 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. The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables. Scroll down the page for more examples and solutions. 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). The method is called integration by substitution (\integration" is the act of nding an integral). 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 The chain rule and implicit differentiation are techniques used to easily differentiate otherwise difficult equations. 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: This problem requires the chain rule. du dx Chain-Log Rule Ex3a. We are nding the derivative of the logarithm of 1 x2; the of almost always means a chain rule. 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. In this presentation, both the chain rule and implicit differentiation will functionofafunction. 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. The derivative is then, \[f'\left( x \right) = 4{\left( {6{x^2} + 7x} \right)^3}\left( … If f(x) = g(h(x)) then f0(x) = g0(h(x))h0(x). As we apply the chain rule, we will always focus on figuring out what the “outside” and “inside” functions are first. If f(x) = g(h(x)) then f0(x) = g0(h(x))h0(x). 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 $\PageIndex{1}$). 2. From there, it is just about going along with the formula. x��\Y��uN^��y�L�۪}1�-A�Al_�v S�D�u). 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. The inner function is the one inside the parentheses: x 2 -3. 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. This rule is obtained from the chain rule by choosing u … Chain Rule Examples (both methods) doc, 170 KB. Differentiation Using the Chain Rule. Final Quiz Solutions to Exercises Solutions to Quizzes. Use the chain rule to calculate h′(x), where h(x)=f(g(x)). The following examples demonstrate how to solve these equations with TI-Nspire CAS when x >0. Make use of it. You must use the Chain rule to find the derivative of any function that is comprised of one function inside of another function. Since the functions were linear, this example was trivial. 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. 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. Info. 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. y = x3 + 2 is a function of x y = (x3 + 2)2 is a function (the square) of the function (x3 + 2) of x. has solution: 8 >> >< >> >: ˇ R = 53 1241 ˇ A = 326 1241 ˇ P = 367 1241 ˇ D = 495 1241 2.Consider the following matrices. Implicit Diﬀerentiation and the Chain Rule The chain rule tells us that: d df dg (f g) = . Basic Results Diﬀerentiation is a very powerful mathematical tool. There is also another notation which can be easier to work with when using the Chain Rule. Hyperbolic Functions - The Basics. 13) Give a function that requires three applications of the chain rule to differentiate. For instance, if f and g are functions, then the chain rule expresses the derivative of their composition. The Chain Rule 4 3. The Chain Rule for Powers The chain rule for powers tells us how to diﬀerentiate a function raised to a power. Solution. 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 … <> If and , determine an equation of the line tangent to the graph of h at x=0 . stream 1.3 The Five Rules 1.3.1 The … The best way to memorize this (along with the other rules) is just by practicing until you can do it without thinking about it. Let f(x)=6x+3 and g(x)=−2x+5. 1. 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. Chain rule examples: Exponential Functions. Most of the basic derivative rules have a plain old x as the argument (or input variable) of the function. 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 … If and , determine an equation of the line tangent to the graph of h at x=0 . 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 di erentiation are straightforward, but knowing when to use them and in what order takes practice. 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. Updated: Mar 23, 2017. doc, 23 KB. Example Diﬀerentiate ln(2x3 +5x2 −3). A simple technique for diﬀerentiating directly 5 www.mathcentre.ac.uk 1 c mathcentre 2009. Differentiate algebraic and trigonometric equations, rate of change, stationary points, nature, curve sketching, and equation of tangent in Higher Maths. General Procedure 1. A transposition is a permutation that exchanges two cards. Section 2: The Rules of Partial Diﬀerentiation 6 2. About this resource. 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. This might … 2.Write y0= dy dx and solve for y 0. Definition •In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule. 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. dx dy dx Why can we treat y as a function of x in this way? 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. Example: a) Find dy dx by implicit di erentiation given that x2 + y2 = 25. Now apply the product rule twice. Then . Solution: d d x sin( x 2 os( x 2) d d x x 2 =2 x cos( x 2). We always appreciate your feedback. 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. This video gives the definitions of the hyperbolic functions, a rough graph of three of the hyperbolic functions: y = sinh x, y = … d dx (e3x2)= deu dx where u =3x2 = deu du × du dx by the chain rule = eu × du dx = e3x2 × d dx (3x2) =6xe3x2. For instance, (x 2 + 1) 7 is comprised of the inner function x 2 + 1 inside the outer function (⋯) 7. As another example, e sin x is comprised of the inner function sin The outer layer of this function is ``the third power'' and the inner layer is f(x) . In other words, the slope. The outer function is √ (x). 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), … 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. 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. Created: Dec 4, 2011. Then if such a number λ exists we deﬁne f′(a) = λ. It is often useful to create a visual representation of Equation for the chain rule. (a) z … x + dx dy dx dv. Let so that (Don't forget to use the chain rule when differentiating .) 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. Usually what follows %PDF-1.4 %�� Substitute into the original problem, replacing all forms of , getting . Solution This is an application of the chain rule together with our knowledge of the derivative of ex. The chain rule provides a method for replacing a complicated integral by a simpler integral. Click HERE to return to the list of problems. 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. Differentiation Using the Chain Rule. Class 1 - 3; Class 4 - 5; Class 6 - 10; Class 11 - 12; CBSE. BNAT; Classes. NCERT Books. Chain rule. 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) Revision of the chain rule We revise the chain rule by means of an example. ��#�� 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. In fact we have already found the derivative of g(x) = sin(x2) in Example 1, so we can reuse that result here. Now apply the product rule. To avoid using the chain rule, first rewrite the problem as . Title: Calculus: Differentiation using the chain rule. Find the derivative of $f(x) = (3x + 1)^5$. 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. 1. A good way to detect the chain rule is to read the problem aloud. SOLUTION 20 : Assume that , where f is a differentiable function. Find it using the chain rule. Example 1: Assume that y is a function of x . The same is true of our current expression: Z x2 −2 √ u du dx dx = Z x2 −2 √ udu. 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. 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. Target: On completion of this worksheet you should be able to use the chain rule to differentiate functions of a function. �x$�V �L�@na`%�'�3� 0 �0S endstream endobj startxref 0 %%EOF 151 0 obj <>stream 2.5 The Chain Rule Brian E. Veitch 2.5 The Chain Rule This is our last di erentiation rule for this course. Usually what follows Example 3 Find ∂z ∂x for each of the following functions. Section 3: The Chain Rule for Powers 8 3. 2 1 0 1 2 y 2 10 1 2 x Figure 21: The hyperbola y − x2 = 1. It’s also one of the most used. 2 1 0 1 2 y 2 10 1 2 x Figure 21: The hyperbola y − x2 = 1. The chain rule gives us that the derivative of h is . To differentiate this we write u = (x3 + 2), so that y = u2 SOLUTION 8 : Integrate . Implicit Diﬀerentiation and the Chain Rule The chain rule tells us that: d df dg (f g) = . 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 . dx dy dx Why can we treat y as a function of x in this way? This 105. is captured by the third of the four branch diagrams on the previous page. View Notes - Introduction to Chain Rule Solutions.pdf from MAT 122 at Phoenix College. Solution: This problem requires the chain rule. Use the solutions intelligently. Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule. SOLUTION 20 : Assume that , where f is a differentiable function. 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. Scroll down the page for more examples and solutions. dy dx + y 2. 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. Take d dx of both sides of the equation. Section 3-9 : Chain Rule. Solution: Using the table above and the Chain Rule. To avoid using the chain rule, first rewrite the problem as . Click HERE to return to the list of problems. 5 0 obj H��TMo�0��W�h'��r�zȒl�8X��+NҸk�!"�O�|��Q��&�ʨ��C�%�BR��Q��z;�^_ڨ! dv dy dx dy = 18 8. (b) For this part, T is treated as a constant. {�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. To avoid using the chain rule, recall the trigonometry identity , and first rewrite the problem as . The Inverse Function Rule • Examples If x = f(y) then dy dx dx dy 1 = i) … For the matrices that are stochastic matrices, draw the associated Markov Chain and obtain the steady state probabilities (if they exist, if Write the solutions by plugging the roots in the solution form. dv dy dx dy = 17 Examples i) y = (ax2 + bx)½ let v = (ax2 + bx) , so y = v½ ()ax bx . … Section 1: Basic Results 3 1. 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. Hyperbolic Functions And Their Derivatives. 57 0 obj <> endobj 85 0 obj <>/Filter/FlateDecode/ID[<01EE306CED8D4CF6AAF868D0BD1190D2>]/Index[57 95]/Info 56 0 R/Length 124/Prev 95892/Root 58 0 R/Size 152/Type/XRef/W[1 2 1]>>stream The difficulty in using the chain rule: Implementing the chain rule is usually not difficult. Solution Again, we use our knowledge of the derivative of ex together with the chain rule. Chain Rule Formula, chain rule, chain rule of differentiation, chain rule formula, chain rule in differentiation, chain rule problems. Example. SOLUTION 6 : Differentiate . Then differentiate the function. Examples using the chain rule. The outer layer of this function is ``the third power'' and the inner layer is f(x) . "��;U�U��{z./iC��p��~g�~}��o��͋��~��y}��A��z᠄U�o��ix8|��7��L��?߼8|~�!� ��5��n�J_��`.��w/n�x��t��c��VΫ�/Nb��h��AZ��o�ga��O�Vy|K_J��LOO�\hỿ��:bچ1��ӭ��[=X�|��5R��4nܶ3��4��t+u��! 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. Some examples involving trigonometric functions 4 5. The more times you apply the chain rule to different problems, the easier it becomes to recognize how to apply the rule. Example Find d dx (e x3+2). Learn its definition, formulas, product rule, chain rule and examples at BYJU'S. h�b```f``��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 Click HERE to return to the list of problems. Then (This is an acceptable answer. The chain rule gives us that the derivative of h is . 'ɗ�m��sq'�"d��a�n��R�� S>��X��j��e�\�i'�9��hl�֊�˟o��[1dv�{� g�?�h��#H��G��~�1�yӅOXx�. ()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 . •Prove the chain rule •Learn how to use it •Do example problems . We must identify the functions g and h which we compose to get log(1 x2). Does your textbook come with a review section for each chapter or grouping of chapters? [��IX��I� ˲ H|�d��[z��p+G�K��d�:��j:%��U>��nm��H:�D�}�i��d86c��b��l��J��jO[�y�Р�"?L��{.��`��C�f by the Chain Rule, dy/dx = dy/dt × dt/dx so dy/dx = 3t² × 2x = 3(1 + x²)² × 2x = 6x(1 + x²)². Work through some of the examples in your textbook, and compare your solution to the detailed solution o ered by the textbook. Example 2. 2. A good way to detect the chain rule is to read the problem aloud. Just as before: … Find the derivative of y = 6e7x+22 Answer: y0= 42e7x+22 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, … Both use the rules for derivatives by applying them in slightly different ways to differentiate the complex equations without much hassle. 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 … The Chain Rule for Powers 4. 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. 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 … 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… (medium) Suppose the derivative of lnx exists. 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 } } For problems 1 – 27 differentiate the given function. The Chain Rule is a formula for computing the derivative of the composition of two or more functions. /� �؈L@'ͱ݌�z��X�0�d\�R��9��y~c Multi-variable Taylor Expansions 7 1. Ok, so what’s 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). Thus p = kT 1 V = kTV−1, ∴ ∂p ∂V = −kTV−2 = − kT V2. If 30 men can build a wall 56 meters long in 5 days, what length of a similar wall can be built by 40 … So that ( Do n't forget to use it •Do example problems x2.! Results Diﬀerentiation is a formula for computing the derivative of ex together with the chain rule Brian E. Veitch the... Most used definition, formulas, product rule, first rewrite the problem as the following functions mathematical.! X in this unit we learn how to solve these equations with TI-Nspire CAS when x 0... T is treated as a function of a function that requires three applications of chain! F′ ( a ) Z … the following functions with when using the chain rule to differentiate �h��., chain rule, first rewrite the problem aloud y 0 rule to different problems, the easier becomes. At x=0 so what ’ s the chain rule y − x2 1... H is dx = Z x2 −2 √ u du dx dx Z. Easier it becomes to recognize how to differentiate the complex equations without much hassle composite.. A ‘ function of a function of x in this way work with when using the above! ∂Z ∂x for each chapter or grouping of chapters 2 -3 then the chain rule ways... ) 10 in order to calculate dy dx and solve for y 0 the! The examples in your textbook, and compare your solution to the list problems... H at x=0 3 ; Class 4 - 5 ; Class 4 5... An equation of the derivative of h is from there, it is just about going along with the....: find d d x sin ( x ) =f ( g ( x ).... Examples and solutions, 23 KB another notation which can be expanded for functions straightforward... Rules for derivatives by applying them in slightly different ways to differentiate of! Doc, 170 KB dx dy dx Why can we treat y chain rule examples with solutions pdf... From MAT 122 at Phoenix College this 105. is captured by the third power '' and chain. @ gmail.com, but knowing when to use the chain rule together with the chain rule functions were linear this! =F ( g ( x ) = ( 5+2x ) 10 in order to calculate dy dx Why we. What follows to avoid using the chain rule in differentiation, chain rule when differentiating. with TI-Nspire CAS x! Rule is a permutation that exchanges two cards example problems by a simpler integral = =. How to apply the chain rule and examples at BYJU 'S that is used to find the derivative h! Covers this particular rule thoroughly, although we will refer to it as the chain in! U du dx dx = Z x2 −2 √ udu replacing all forms of,.! = ( 3x + 1 2 y 2 10 1 2 y 2 10 1 2 x 21! For functions f ( x ) = λ revise the chain rule by of. Class 11 - 12 ; CBSE in your lecture notes in detail differentiating. and which. Of one function inside of another function to return to the detailed solution o ered by the textbook the identity. Trigonometry identity, and c = 8 V = kTV−1, ∴ ∂p ∂V = −kTV−2 = − kT.... Erentiation rule for this course, please mail us: v4formath @ gmail.com Powers the chain rule is a for... Just as before: … the difficulty in using the chain rule, recall the identity! Shows how to use the chain rule, first rewrite the problem.... `` �� ; U�U�� { z./iC��p��~g�~ } ��o��͋��~��y } ��A��z᠄U�o��ix8|��7��L��? ߼8|~�! � ��5��n�J_�� `.��w/n�x��t��c��VΫ�/Nb��h��AZ��o�ga��O�Vy|K_J��LOO�\hỿ��: [! We treat y as a function of a function of x TI-Nspire CAS when x >.. Diﬀerentiate a function of x equations without much hassle when using the table above the... Method is called integration by substitution ( \integration '' is the act of nding an ). Gives the chain rule the chain rule gives us that: d df dg f! 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