Chain rule questions

Solution: This is a composition of three functions f(g(h(x))), where h(x) = x5 1, g(x) = p x and f(x) = sin(x). Let u = ax + b, then y = u½. Chain Rule revision and practice questions. Mar 24, 2023 · In Chain Rule for One Independent Variable, the left-hand side of the formula for the derivative is not a partial derivative, but in Chain Rule for Two Independent Variables it is. 8 Derivatives of Hyperbolic Functions; 3. by the Chain Rule, dy/dx = dy/dt × dt/dx. 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 1). The product rule. How many hours a day must 4 pumps work to empty the tank in 1 day? 9. AffairsCloud Today. Save 20 questions What the questions look like. Afterwards, you take the derivative of the inside part and multiply that with the part you found previously. 3 Volumes of Solids of Revolution / Method of Nov 3, 2015 · Questions Regarding this Chain Rule Proof. Then for block two, d/dx he had to use the product rule to expand it. The chain rule starts with a composite function f(g(x)). The quotient rule. S. Learn. A heated metal ball is dropped into a liquid. Worked example: Derivative of sec (3π/2-x) using the chain rule. Reverse chain rule. For example, if g = s(x, y) + it(x, y) g = s ( x, y) + i t ( x, y) we have that the term ∂g ∂z = ∂g ∂x − i∂g The Chain Rule for Functions of Several Variables 7 Elsewhere we saw how to compute dy dx when y is an implicit function of x. P. function. For such problems, the chain rule and the chain rule formula is very effective. Derivative of aˣ (for any positive base a) Derivative of logₐx (for any positive base a≠1) Worked example: Derivative of 7^ (x²-x) using the chain rule. What we're gonna do in this video is focus on key misunderstandings that folks often have, and we actually got these misunderstandings from the folks who write the AP exams, from the actual College Board. He uses the product rule to put it into proper form and solves. = 400 e–0. composite function. Hence, the Nov 16, 2022 · The chain rule for this case is, dz dt = ∂f ∂x dx dt + ∂f ∂y dy dt d z d t = ∂ f ∂ x d x d t + ∂ f ∂ y d y d t. y= (x^2+3x-1)^2 y = (x2 +3x−1)2. By breaking down the function into its components, sqrt (x) and 3x^2-x, we demonstrate how their derivatives work together to make differentiation easier. The chain rule states formally that. In the table below, observe how each basic derivative in the rst column (based on material through Section 3. The inner function is u=x^2+3x-1 u = x2 +3x −1. \(F_1(x) = (1-x)^2\): Feb 16, 2023 · The chain rule states that the derivative of f(g(x)) is f'(g(x))⋅g'(x). Login to view the problems and chat with a tutor if you need additional help. 12 men working 8 hours per day complete a piece of work in 10 days. What is f' (x) if f (x) = cos (5x4)? Chain Rule Practice quiz for 11th grade students. So, basically what we’re doing here is differentiating f f with respect to each variable in it and then multiplying each of these by the derivative of that variable with respect to t t. STEP 3: Integrate and simplify. For example, sin(x²) is a composite function because it can be constructed as f(g(x)) for f(x)=sin(x) and g(x)=x². 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. Note that for #1 the first block he solves for the regular derivative. Find dy dx. 3. Derivatives of a composition of functions, derivatives of secants and cosecants. Multivariable Chain Rules allow us to differentiate z with respect to any of the variables involved: Let x = x(t) and y = y(t) be differentiable at t and suppose that z = f(x, y) is differentiable at the point (x(t), y(t)). 10. ( 2 x 2 + 2 π) d x . The chain rule is used to di it. Find ∂w ∂r ∂ w ∂ r and ∂w ∂s ∂ w ∂ s. 6) (3. 2. Then differentiate the. STEP 2: ‘Adjust’ and ‘compensate’ any numbers/constants required in the integral. In school, there are some chocolates for 240 adults and 400 children. 6 Definition of the Definite Integral; 5. Differentiate y = 3√1 −8z y = 1 − 8 z 3 . The chain rule. Created by Sal Khan. Online exams, practice questions and revision videos for every GCSE level 9-1 topic! No fees, no trial period, just totally free access to the UK’s best GCSE maths revision platform. 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. To avoid confusion, we ignore most of the subscripts here. 2 Area Between Curves; 6. Use the Chain Rule to find the derivatives of the following functions, as given in Example 59. No videos or articles available in this lesson; Practice. Some of the types of chain rule problems that are asked in the exam. View Answer. Importance:Medium. ∫ 1 x log x dx = ∫ 1 log x ⋅ 1 x dx = ∫ 1 u du ∫ 1 x log x d x = ∫ 1 log x ⋅ 1 x d To do the chain rule you first take the derivative of the outside as if you would normally (disregarding the inner parts), then you add the inside back into the derivative of the outside. Dec 29, 2020 · Alternate Chain Rule Notation; We have covered almost all of the derivative rules that deal with combinations of two (or more) functions. 11. However, we rarely use this formal approach when applying the chain 4. Critical thinking question: 13) Give a function that requires three applications of the chain rule to differentiate. This diagram can be expanded for functions of more than one variable, as we shall see Dec 21, 2020 · It will take a bit of practice to make the use of the chain rule come naturally---it is more complicated than the earlier differentiation rules we have seen. Find the temperature of the ball as it enters the liquid. Example 3. By applying the chain rule, we illuminate the process, making it easy to understand. 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 13. Let's use some notation. 7 Derivatives of Inverse Trig Functions; 3. The chain rule is a method for determining the derivative of a function based on its dependent variables. A special rule, the chain rule, exists for differentiating a function of another function. 1 Average Function Value; 6. Find the derivative of the following function with respect to x x. Listed below are a few solved examples on Chain Rule to enhance your understanding of the concept: Chain Rule Example 1: Solve, y (x) = (2x2+ 8)2. And finally multiplies the result of the first chain rule application to the result of the second chain rule application. If the chocolates are taken away by 300 children, then how many adults will be provided with the remaining chocolates? The Chain Rule formula is a formula for computing the derivative of the composition of two or more functions. Jun 15, 2022 · The chain rule is the method for computing the derivative of a composite function. He then goes on to apply the chain rule a second time to what is inside the parentheses of the original expression. For the sake of clarity g(f(x)) would be sin^2(x). Through a worked example, we explore the Chain rule with a table. Furthermore, the product rule, the quotient rule, and the chain rule all hold for such complex functions. 5 Quotient Rule for the Edexcel A Level Maths: Pure syllabus, written by the Maths experts at Save My Exams. The chain rule applied to the function sin(x) and p x5 1 gives Practice Online IB Style Practice Questions for IB Math Analysis & Approaches -Topic: SL 5. Monthly Current Affairs. Dec 19, 2023 · Chain Rule Questions are important topics for anyone who is preparing for upcoming govt exams. Government Jobs 2024 Notification. Most problems are average. 100+ Chain rule problems for Bank exams - 1. Chain Rule - General Questions. Mark Scheme. Next. So let's say that we are trying to take the derivative of the expression. Multiple Choice. Part 1 explanation: 1. For questions involving the chain rule in analysis. 6 The chain rule for composite functions SL Paper 1 About Transcript. Anyway, the chain rule says if you take the derivative with respect to x of f(g(x)) you get f Feb 15, 2024 · The Chain Rule: Example 1. Applications of Integrals. Such as sin(x^2), where one function is the sine operation and the other is the squared operation. In function notation, if then the chain rule can be written as: Question 7: What is the origin of chain rule? Answer: The chain rule has been in existence since the discovery of calculus by Isaac Newton and Leibniz at the end of the 17th century. The integrals requiring the Reverse Chain Rule technique are identified by having. Hot Network Questions Mad scientist spoof - identical snowflakes Questions involving the chain rule will appear on homework, at least one Term Test and on the Final Exam. Oct 23, 2017 · When you transpose a vector. Answer. The reason is that, in Chain Rule for One Independent Variable, \(z\) is ultimately a function of \(t\) alone, whereas in Chain Rule for Two Independent Variables The chain rule is a rule for differentiating compositions of functions. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you. Also learn how to use all the different derivative rules together in a thoughtful and strategic manner. Past paper questions for the Chain Rule topic of A-Level Edexcel Maths. Learn how to differentiate composite functions using the chain rule, and practice with examples and exercises. This is a chain rule, within a chain rule problem. f (x) = ln (sin (3x^2 - 5)) View Answer. then the transpose is. After reading this text, and/or viewing The Chain Rule tells us how! Example: Sage the Dog can run 3 times faster than you, and you can run 2 times faster than me, so Sage can run 3 × 2 = 6 times faster than me. It can be shown that ƒ is holomorphic, and that ƒ'(z) = 1 - 3𝑖 for every complex number z . Chain Rule - Formulas. Chain rule for function of several variables. 2) Indirect Proportion: Any two quantities are said to be indirectly proportional, if on the increase of one quantity, the other quantity decreases and vice-versa. 7 Computing Definite Integrals; 5. (A) 4. Then du = 1 x dx d u = 1 x d x. 6. A tree diagram can be used to represent the dependence of variables on other variables. Curriculum-based maths in NSW. zT = (Ax)T =xTAT. First, a composite function takes the form: F (x) = f (g (x)) or F = f \circ g. 1) y = 4x5(4x4 − 2) 2) y = −2x5(2x2 − 3) Applying the product rule is the easy part. dT dt = ∂T ∂x dx dt + ∂T ∂y dy dt. Chain Rule quiz for 12th grade students. df dt = ∂f ∂x dx dt + ∂f ∂ydy dt. Jun 14, 2019 · The Chain Rule for Functions of Multiple Variables (Exercises) In exercises 1 - 6, use the information provided to solve the problem. Find dw dt d w d t. Bank Jobs 2024 Notification. The Chain Rule is a powerful tool for differentiating functions that involve one function inside of another function. Chain rule is used to find out this missing part of an element by subsequent comparison. This computation is quite tedious to verify and we end up with about 32 terms that needs to cancel out. It is often useful to create a visual representation of Equation 13. Sometimes for the complex type of functions, finding the derivative is very hard. The chain rule states that the derivative of f (g (x)) is f' (g (x))⋅g' (x). In more awkward cases it can help to write the numbers in before integrating. 2) Let w(t, v) = etv w ( t, v) = e t v where t = r + s t = r + s and v = rs v = r s. Whether we are finding the equation of the tangent line to a curve, the instantaneous velocity of a moving particle, or the instantaneous rate of change of a certain quantity, if the function under consideration is a composition, the chain rule is indispensable. Let's dive into the process of differentiating a composite function, specifically f (x)=sqrt (3x^2-x), using the chain rule. Example 59 ended with the recognition that each of the given functions was actually a composition of functions. How to use the chain rule for derivatives. The Reverse Chain Rule is the most common format for the functions we encounter - especially at first. 5 Derivatives of Trig Functions; 3. A more general chain rule. A few are somewhat challenging. The first line is better for understanding and more natural, the second line is a burden from the vector analysis to have the gradient to be a vector instead of a row. It states that for functions f (x) and g (x), (f∘g)′ (x)=f′ (g (x))g′ (x). Chain Rule. In differential calculus, the chain rule is a formula used to find the derivative of a composite function. As for why the chain rule is valid well that's a completely different issue, which you should probably ask in a separate question if this Differentiation is the process through which we can find the rate of change of a dependent variable in relation to a change of the independent variable. The single variable chain rule tells you how to take the derivative of the composition of two functions: d d t f ( g ( t)) = d f d g d g d t = f ′ ( g ( t)) g ′ ( t) The chain rule says (f(g(x)))0= f0(g(x))g0(x), or (f(u))0= f0(u)u0(x) if u = g(x). Difficulty level:1 out of 5 (Easy) Examination-SSC, Banking, GMAT, GRE, SAT, CAT, etc. In single-variable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. z = f(x, y) = x2 − 3xy + 2y2, x = x(t) = 3sin2t, y = y(t) = 4cos2t. 9 Chain Rule; 3. In other words, it helps us differentiate *composite functions*. (indicated by red font in the parenthesis) 3. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. 20 interactive practice Problems worked out step by step Learn. Use the chain rule to find the derivative of f (x)= (3x - 1/x^2 + 3)^2. Proving the chain rule for derivatives. 05t + 25, t ≥ 0. Substitute u = ax + b back into your answer. The chain rule tells us how to find the derivative of a composite function. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. This unit illustrates this rule. For example, sin (x²) is a composite function because it can be constructed as f (g (x)) for f (x)=sin (x) and g (x)=x². Transcript. Video transcript. 5. Practicing these questions will improve your calculation skills will help you to solve the questions in competitive exams. 3 pumps, working 8 hours a day, can empty a tank in 2 days. aptitude. Find other quizzes for Mathematics and more on Quizizz for free! 5 days ago · Chain Rule Derivative Solved Examples. In other words, the differential of something in a bracket raised to the power Chain Rule - Learn and practice Chain Rule with solved Aptitude Questions and Answers accompanied by easy explanation, shortcuts and tricks that help in understanding the concept clearly. The chain rule now adds substantially to our ability to compute derivatives. Aptitude Questions : Chain Rule Set 1. Hint. 6) can be generalized by the chain rule by using g(x) in place of x in Higher; Differentiation The chain rule. 6 Derivatives of Exponential and Logarithm Functions; 3. In the following discussion and solutions the derivative of a function h ( x) will be denoted by or h ' ( x) . As an example, consider the function ƒ: C → C defined by ƒ(z) = (1 - 3𝑖)z - 2 . The operations of addition, subtraction, multiplication (including by a constant) and division led to the Sum and Difference rules, the Constant Multiple Rule, the Power Rule, the Product Rule and the Quotient Rule. Nov 16, 2022 · Determine f uu f u u for the following situation. Worked example: Derivative of log₄ (x²+x) using the chain rule. Parent section:Quantitative Aptitude. To carry out the chain rule, know basic derivatives well so you can build on that. 4 Product and Quotient Rule; 3. f = f (x,y) x = u2 +3v, y = uv f = f ( x, y) x = u 2 + 3 v, y = u v Solution. Find h ′ ( x) . May Current Affairs 2024. 0. 6) 625 − x 2. The Chain Rule is a tool for differentiating a composite for functions. 1 for the chain rule. So it can be written as f (g (x)). Suppose that z = f(x, y), where x and y themselves depend on one or more variables. Then he combines then using distribution. 11 and 4. Jul 25, 2020 · And yes, your chain rule tree looks right (that's how you can get 4. Use Equation 10. To put this rule into context, let’s take a look at an example: h(x) = sin(x3). Differentiate both parts separately. This method can be used for any fractional power of any linear or non-linear expression. There are many formats for functions which we need to integrate. AC Team 1 - December 5, 2015. z = Ax z = A x. Question 1 . x. STEP 1: Spot the ‘main’ function. MME gives you access to maths practice questions, worksheets and videos. Earlier in the class, wasn't there the distinction between Chain Rule Examples. Next substitute u= (x^2 + 1)^3, meaning du/dx = 6x (x^2 + 1)^3. Solution. Chain rule in differentiation is defined for composite functions. = 6x (1 + x²)². Back to the integral: By substitution, we get. So to continue the example: d/dx[(x+1)^2] 1. I am. Current Affairs APP. Home. In its simplest form, it says that if f(x, y) is a function of two variables and x(t) and y(t) depend on t, then. Such questions may also involve additional material that we have not yet studied, such as higher-order derivatives. In how many days will 30 persons, working 6 hours a day, complete the work? If 20 men can build a wall 56 meters long in 6 days , what length of a similar wall can be built by View Answer. 3 Substitution Rule for Indefinite Integrals; 5. Proving the chain rule. 5 Area Problem; 5. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. t. 10 Implicit Differentiation; 3. Chain Rule quiz for 11th grade students. In the next example we consider an alternative (and quicker) method using the chain rule as formulated in Example 3. You can also create similar chain-rule trees for $\frac{\partial u}{\partial \xi}$ and $\frac{\partial u}{\partial \eta}$. For example, the chain rule cannot be used to take the derivative of x^x. Revision notes on 7. g. one part of their expression being the derivative of the other part. Google Classroom. Chain rule with tables Get 3 of 4 questions to level up! Review: Product, quotient, & chain rule. As the ball cools, its temperature, T °C, minutes after it enters the liquid, is given by. While you can treat "2" as a function, namely the constant function f(x) = 2 that outputs 2 for all inputs, raising one function to the power of another like f(x)^g(x) is different from composing functions, as in f(g(x)). Chain rule. z T = ( A x) T = x T A T. . This is not a product rule problem. You will also see chain rule in MAT 244 (Ordinary Differential Equations) and APM 346 (Partial Differential Equations). Using specific x-values for functions f and g, and their derivatives, we collaboratively evaluate the derivative of a composite function F (x) = f (g (x)). This is an exceptionally useful rule, as it opens up a whole world of functions (and equations!) we can now differentiate. For instance, if f and g are functions, then the chain rule expresses the derivative of their composition. Find other quizzes for Mathematics and more on Quizizz for free! Furthermore, the product rule, the quotient rule, and the chain rule all hold for such complex functions. 1 to explain why the instantaneous rate of change of T that results from a change in t is. To find the derivative of log_e (x^2+1)^3 use chain rule. See the following definition: Why is the chain rule called "chain rule". Differentiate composite functions (all function types) ( 1 x) . The more questions a candidate answers, the better their understanding of the Chain Rule concepts and their ability to answer quickly and accurately. Put both parts into the chain rule. Nov 10, 2020 · Example 60: Using the Chain Rule. Nov 16, 2022 · Section 3. Quotient rule from product & chain rules. Share. June Current Affairs 2024. Year 11 Maths Advanced. If y = f (g (x)), then as per chain rule the instantaneous rate of change of function ‘f’ relative to ‘g’ and ‘g’ relative to x results in an instantaneous rate of change of ‘f’ with respect to ‘x’. Question 1. The other given part of the same element is taken as base and is compared separately with all the other elements e. 1️⃣ Define your inner and outer functions. Show Solution. To complete the same work in 8 days, working 15 hours a day, the number of men required is. 11 Related Rates; 3. Use the Chain Rule to find partialz/partials and partialz/partialt. The reason is that we can chain even more functions together. 3. 12 Higher Order Derivatives; 3. Here is a set of practice problems to accompany the Chain Rule section of the Partial Derivatives chapter of the notes for Paul Dawkins Calculus III course at Lamar University. 12). Worked example: Derivative of ∜ (x³+4x²+7 Exercise : Chain Rule - General Questions. Copy & Edit. Worked example: Quotient rule with table. Rewrite. Learn how to use the Chain Rule to find derivatives of composite functions, with examples, hints and practice problems. 13 Logarithmic Differentiation; 4 The chain rule is a formula that allows you to differentiate composite functions. About. Back to Problem List. z = 3 sin theta cos phi, theta = st^6, phi = s^5t. and. This essentially means that F (x) is a composite function, where f (x) is the outer function and g (x) is the inner function. Question: 1. 4 More Substitution Rule; 5. The chain rule is a special rule to differentiate a composition (chain) of several functions. The chain rule tells us how to find the derivative of a composite function: d d x [ f ( g ( x))] = f ′ ( g ( x)) g ′ ( x) The AP Calculus course doesn't require knowing the proof of this rule, but we believe that as long as a proof is accessible, there's Jun 26, 2023 · Use known derivative rules, including the chain rule, as needed to answer each of the following questions. Example 4 Using the Chain Rule for Implicit Differentiation Suppose 3xy - exy = 5. In this blog, we have mentioned 15 chain rules questions and answers so you can practice and do self evaluation. mc-TY-chain-2009-1. so dy/dx = 3t² × 2x = 3 (1 + x²)² × 2x. Sep 7, 2022 · Instead, we use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Find other quizzes for Mathematics and more on Quizizz for free! 20 questions. Chain Rule Questions. If 7 spiders make 7 webs in 7 days, then 1 spider will make 1 web in how many days? 39 persons can repair a road in 12 days, working 5 hours a day. Nov 16, 2022 · Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule. Using the chain rule and the derivatives of sin (x) and x², we can Aug 6, 2014 · 2 Answers. Find topic revision quizzes, diagnostic quizzes, extended response questions, past papers, videos and worked solutions for Chain Rule. 2 3. The chain rule says that the derivative of a composite function is the inner function multiplied by the derivative of the outer function. Practice Chain Rule Questions & Answers. You will often find many cases like expoential, trigonmetric, logarithmic, inverse trigonometric expressions in which you need to use chain rule so can find the derivative so you need to be comfortable with it. Questions. In function notation, if then the chain rule can be written as: The Chain Rule. 8 Substitution Rule for Definite Integrals; 6. 625 −x2− −−−−−−√. The rule remains the same, you just have to do it twice: differentiate the outermost function, keep the inside the same, then multiply by the derivative of the inside. If z is a function of y and y is a function of x, then the derivative of z with respect to x can be written \frac{dz}{dx} = \frac{dz}{dy}\frac{dy}{dx}. Evaluate the derivative of the function given below. if you have a question having following elements: ‘men’, ‘days’ and ‘hours’. Quotient rule review. 9 : Chain Rule. In calculus, the chain rule is a formula used to find the derivative of a compound The chain rule will allow us to find the derivative of this. Find the value of t for which T = 300, giving your answer to 3 significant figures. am assuming familiarity with the chain rule. If y is a function of u, and u is a function of x, then the chain rule tells us that: Note that because u is a function of x, y is also a function of x. Example: Let us compute the derivative of sin(p x5 1) for example. (3. exercise-3. Be careful - the only multiplication going on in that problem is the "ax" part. Correspondingly, the RHS must change. The rule facilitates calculations that deal with derivatives of complex expressions. Aug 29, 2023 · In the Chain Rule you can think of the function in question as the composition of an “outer” function \(f\) and an “inner” function \(u\); first take the derivative of the “outer” function then multiply by the derivative of the “inner” function. Solve five chain rule problems with step-by-step solutions. This rule will help us find the derivative of F (x). Many answers: Ex. The outer function is y=u^2 y = u2. 2️⃣ Take the derivative of the outer function only! Multi-Variable Chain Rule. 1) Direct Proportion: Any two quantities are said to be directly proportional, if on the increase of one quantity, the other quantity increases and vice-versa. We are given the definition that ∂ ∂z = 1 2( ∂ ∂x − i ∂ ∂y) ∂ ∂ z = 1 2 ( ∂ ∂ x − i ∂ ∂ y). A composite function is a function h (x) formed by using the output of one function g (x) as the input of another function f (x). . The chain rule is a formula that allows you to differentiate composite functions. As you can probably imagine, the multivariable chain rule generalizes the chain rule from single variable calculus. Sep 29, 2023 · The Chain Rule tells us about the instantaneous rate of change of T, and this can be found as. lim Δt → 0ΔT Δt = lim Δt → 0TxΔx + TyΔy Δt. y = f (u) u = f (x) dy = dy ⋅ du dx du dx. The chain rule allows us to use substitution to differentiate any function in the form. We need to determine du d u in order to take into account (reverse, so to speak) the use of the chain rule involved in differentiating the desired function. 1. 1 ). Nov 16, 2022 · 5. Differentiation of algebraic and trigonometric expressions can be used for calculating rates of change, stationary points and their nature, or the gradient 4. Solution: Here, as you can see that y (x) is a composite function. Integrating with reverse chain rule. We already know that the answer is −x/ 625 −x2− −−− Product Rule and Chain Rule Practice Differentiate each function with respect to x. It is often useful to create a visual representation of the Chain Rule for One Independent Variable for the chain rule. Compute the derivative of. Very useful for all freshers, college students and engineering students preparing for placement tests or any competitive exam like MBA, CAT, MAT, SNAP, MHCET Nov 16, 2022 · 3. Find an equation for the tangent line to the curve \(y = \sqrt{ e^x + 3}\) at the point where \(x = 0\). Tangent to y=𝑒ˣ/ (2+x³) Normal to y=𝑒ˣ/x². iu lz mu qw mb nm yd ll zg vo