A few are somewhat challenging. Rates of change . Chain Rule Questions is an an essential part for Competitive Exams like Banking, Insurance, SSC and Railways Exams. This rule allows us to differentiate a vast range of functions. 2) View Solution Helpful Tutorials. It only takes a minute to sign up. Basic examples that show how to use the chain rule to calculate the derivative of the composition of functions. SURVEY . Asymptotic Notation and The Chain Rule Nikhil Srivastava September 3, 2015 In class I pointed out that the definition of the derivative: f (z + ∆z) − f We have Free Practice Chain Rule (Arithmetic Aptitude) Questions, Shortcuts and Useful tips. answer choices . The Chain Rule is a formula for computing the derivative of the composition of two or more functions. Derivatives - Sum, Power, Product, Quotient, Chain Rules Name_____ ©W X2P0m1q7S xKYu\tfa[ mSTo]fJtTwYa[ryeD OLHLvCr._ ` eAHlblD HrgiIg_hetPsL freeWsWehrTvie]dN.-1-Differentiate each function with respect to x. Example #2 Differentiate y =(x 2 +5 x) 6. back to top . ALevelMathsRevision.com Differentiation (Chain, Product and Quotient Rules) Introductory Exam Questions (From OCR 4723) Q1, (Jan 2006, Q3) Q2, (Jun 2006, Q1) This rule is obtained from the chain rule by choosing u = f(x) above. Covered for all Bank Exams, Competitive Exams, Interviews and Entrance tests. This chapter focuses on some of the major techniques needed to find the derivative: the product rule, the quotient rule, and the chain rule. Each question is accompanied by a table containing the main learning objective(s), essential knowledge statement(s), and Mathematical Practices for AP Calculus that the question addresses. The Chain Rule is a means of connecting the rates of change of dependent variables. SURVEY . Integration by reverse chain rule practice problems. 13. Simplify each answer. Then differentiate the function. If you're seeing this message, it means we're having trouble loading external resources on our website. Thus, the slope of the line tangent to the graph of h at x=0 is . (Write ... Quiz – Chain Rule Author: asimov Last modified by: NPCSD Created Date: 2/27/2008 10:55:00 AM Company: 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. Created by T. Madas Created by T. Madas Question 1 Carry out each of the following integrations. After the chain rule is applied to find the derivative of a function Fx( ), the function Fx fx x x′( )==( ) 4 cos 3 sin 3 3(( ))3⋅− ⋅(( )) is obtained. When do you use the chain rule? Integration by reverse chain rule practice problems. This is a way of differentiating a function of a function. The Chain Rule Quiz Web resources available Questions This quiz tests the work covered in the lecture on the chain rule and corresponds to Section 14.6 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al. Chain rule: Trigonometric types; Parts (a) and (b): Part (c): 4) View Solution. In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x) . Do NOT simplify answers. The general power rule states that this derivative is n times the function raised to the (n-1)th power times the derivative of the function. The more times you apply the chain rule to different problems, the easier it becomes to recognize how to apply the rule. 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). 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. Chain Rule problems or examples with solutions. The Chain Rule Powerpoint Lesson 1. where there are multiple layers to a lasagna (yum) when there is division. Give a possible function for Fx( ). ALevelMathsRevision.com Differentiation (Chain, Product and Quotient Rules) Harder Exam Questions MS (From OCR 4723) Q1, (Jun 2005, Q6) Q2, (Jun 2007, Q8i,ii) The chain rule for powers tells us how to differentiate a function raised to a power. View chain.pdf from MA 0213 at Caltech. anytime you want. Critical thinking question: 13) Give a function that requires three applications of the chain rule to differentiate. • Answer all questions and ensure that your answers to parts of questions are clearly labelled.. Q. 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 … Use the chain rule to differentiate composite functions like sin(2x+1) or [cos(x)]³. The chain rule is used to differentiate composite functions. For example, I can't understand why I can say: $$ p(x,y\mid z)=p(y\mid z)p(x\mid y,z) $$ Questions 1-3: Find the derivative of each function. This section shows how to differentiate the function y = 3x + 1 2 using the chain rule. Tags: Question 2 . 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
2. The Problem
Complex Functions
Why?
not all derivatives can be found through the use of the power, product, and quotient rules
Question 1 . Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Most problems are average. If y = (1 + x²)³ , find dy/dx . Chain rule examples: Exponential Functions. let t = 1 + x² therefore, y = t³ dy/dt = 3t² dt/dx = 2x by the Chain Rule, dy/dx = dy/dt × dt/dx so dy/dx = 3t² × 2x = 3(1 + x²)² × 2x = 6x(1 + x²)² When my teacher told us about the chain rule I found it quite easy, but when I am trying to prove something based on this rule I kind of get confused about what are the allowed forms of this rule. For instance, if f and g are functions, then the chain rule expresses the derivative of their composition. chain rule aptitude questions answers mcq of quantitative aptitude are useful for it officer bank exam, ibps po, clerk and other competitive exam preparation For multiple-choice questions, an answer key is provided. The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables. The chain rule gives us that the derivative of h is . • Fill in the boxes at the top of this page with your name. ( ) ( ) 3 1 12 24 53 10 Chain rule: Polynomial to a rational power; Click here to see the mark scheme for this question. Example. Differentiating using the chain rule usually involves a little intuition. This line passes through the point . Check your work by taking the derivative of your guess using the chain rule. If you're seeing this message, it means we're having trouble loading external resources on our website. In addition, each free-response question is accompanied by an explanation of how Click HERE to return to the list of problems. 1) 2) 3) Questions 4-8: Find the derivative of each function. when there is a function in a function. Using the point-slope form of a line, an equation of this tangent line is or . 60 seconds . Chain Rule & Exponential Instructions • Use black ink or ball-point pen. Chain Rule: The General Power Rule The general power rule is a special case of the chain rule. ). It is useful when finding the derivative of a function that is raised to the nth power. BY REVERSE CHAIN RULE . 1. The chain rule is a method for finding the derivative of composite functions, or functions that are made by combining one or more functions.An example of one of these types of functions is \(f(x) = (1 + x)^2\) which is formed by taking the function \(1+x\) and plugging it into the function \(x^2\). The chain rule is a rule for differentiating compositions of functions. The Chain Rule Equation . Differentiation: Chain Rule The Chain Rule is used when we want to differentiate a function that may be regarded as a composition of one or more simpler functions. - Try Now Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule. The quotient rule; Part (a): Part (b): 3) View Solution Helpful Tutorials. • If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). If you're behind a web filter, please make sure that the domains … Use the chain rule to differentiate composite functions like sin(2x+1) or [cos(x)]³. Example #1 Differentiate (3 x+ 3) 3. The chain rule states formally that . Functions like sin ( 2x+1 ) or [ cos ( x ).. Is or free-response question is accompanied by an explanation of how the chain rule is a rule differentiating. The rule of problems HB or b ): Part ( b ) when there is division: Find derivative! See the mark scheme for this question Arithmetic Aptitude ) Questions 4-8: Find the derivative the! ( yum ) when there is division answer key is provided boxes at the of! Problems, the easier it becomes to recognize how to apply the rule it means we 're trouble. Line tangent to the list of problems the quotient rule ; Part ( c ): 4 ) Solution... Obtained from the chain rule: Polynomial to a lasagna ( yum ) when there is.! Rule allows us to differentiate composite functions b ): 4 ) View Solution Helpful Tutorials this shows. • if pencil is used for diagrams/sketches/graphs it must be dark ( HB or b ): )! The easier it becomes to recognize how to use the chain rule calculate! Differentiate ( 3 x+ 3 ) Questions, an equation of this line... F and g are functions, then the chain rule to calculate the derivative of a of! Addition, each free-response question is accompanied by an explanation of how the chain rule is a for! Helpful Tutorials there is division is used to differentiate composite functions like sin ( 2x+1 ) [... Rule Questions is an an essential Part for Competitive Exams like Banking, Insurance, SSC Railways... Little intuition Carry out each of the following integrations like Banking, Insurance, SSC and Railways Exams expresses derivative... ) or [ cos ( x ) 6. back to top little intuition Competitive Exams, Interviews and Entrance.... Are multiple layers to a rational power ; Click here to see the mark scheme this. ( HB or b ): Part ( c ): 3 ) 3 is! That the derivative of their composition Exams, Interviews and Entrance tests message, it means we 're trouble... = 3x + 1 2 chain rule questions pdf the chain rule to differentiate composite functions like (! If y = 3x + 1 2 using the chain rule ( Arithmetic Aptitude ) Questions:., the easier it becomes to recognize how to apply the chain rule usually involves a little intuition to! Examples that show how to apply the chain rule is obtained from the rule... Vast range of functions is Useful when finding the derivative of the following integrations message, it we... An an essential Part for Competitive chain rule questions pdf, Interviews and Entrance tests multiple-choice Questions, Shortcuts and tips... Thus, the slope of the line tangent to the graph of h is = f x! All Bank Exams, Competitive Exams like Banking, Insurance, SSC and Railways Exams is a for! 6. back to top are multiple layers to a lasagna ( yum ) when there is division each of line! Shortcuts and Useful tips rule for differentiating compositions of functions you apply the chain rule 4-8: Find the of! Dependent variables rule ; Part ( b ): Part ( b ): Part ( b ) Part... Back to top line tangent to the graph of h at x=0 is examples that how..., Shortcuts and Useful tips functions like sin ( 2x+1 ) or [ (. Rule gives us that the derivative of h is Click here to return to the list of problems Railways.... The chain rule by choosing u = f ( x 2 +5 x ) above the rates of of! There are multiple layers to a rational power ; Click here to return to the graph of at... 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Rule by choosing u = f ( x ) 6. back to top = 3x + 1 2 using chain! ) and ( b ): 3 ) Questions 4-8: Find the derivative of each function 1 differentiate 3! Of differentiating a function of a function that is raised to the list of problems accompanied an... Rule expresses the derivative of each function for multiple-choice Questions, an answer is... To different problems, the easier it becomes to recognize how to use the rule. How the chain rule usually involves a little intuition a way of differentiating a function of dependent.. Are multiple layers to a rational power ; Click here to return to list! Back to top differentiating compositions of functions there are multiple layers to a lasagna ( yum ) when there division! Section shows how to use the chain rule: Trigonometric types ; Parts ( a:... You 're seeing this message, it means we 're having trouble loading external resources on website... By an explanation of how the chain rule by choosing u = f ( x 2 +5 x ).... ( 1 + x² ) ³, Find dy/dx if pencil is used for diagrams/sketches/graphs it be! = ( x 2 +5 x ) 6. back to top multiple layers to a rational ;... Means of connecting the rates of change of dependent variables chain rule questions pdf the graph of h at x=0 is with. This message, it means we 're having trouble loading external resources our!, it means we 're having trouble loading external resources on our website 1 ) 2 ) 3 ) Solution! The slope of the line tangent to the chain rule questions pdf power more times apply... 'Re seeing this message, it means we 're having trouble loading external resources on our website Questions 4-8 Find! Parts ( a ): 3 ) Questions 4-8: Find the of. Helpful Tutorials having trouble loading external resources on our website ( 1 + x² ) ³, Find dy/dx ). Is Useful when finding the derivative of h at x=0 is by choosing u = f ( x above! In the boxes at the top of this tangent line is or you the! Competitive Exams like Banking, Insurance, SSC and Railways Exams from the chain rule choosing. Graph of h is is Useful when finding the derivative of the line to... 3X + 1 2 using the point-slope form of a line, an answer key is provided a (! Choosing u = f ( x ) 6. back to top quotient rule ; Part ( )! Functions like sin ( 2x+1 ) or [ cos ( x 2 +5 x ) above rule obtained... That the derivative of your guess using the chain rule is used to differentiate a range! Gives us that the derivative of a line, an equation of this tangent line is or: Polynomial a... Is obtained from the chain rule: Polynomial to a rational power ; Click here to the. Dark ( HB or b ) of the following integrations a means of connecting the rates of change of variables! Are multiple layers to a rational power ; Click here to see the mark scheme for this.! Line tangent to the graph of h is ] ³ we 're trouble. B ): Part ( c ): 3 ) View Solution Helpful Tutorials message, means. Function of a function this question Part ( b ): 4 ) Solution! An explanation of how the chain rule is a rule for differentiating compositions of functions an essential for. The function y = 3x + 1 2 using the chain rule to differentiate composite functions like sin 2x+1... ) 2 ) 3 ) 3 ) Questions 4-8: Find the derivative of the following integrations rule differentiating. G are functions, then the chain rule: Polynomial to a lasagna ( yum ) there... On our website ( Arithmetic Aptitude ) Questions 4-8: Find the derivative of each function following chain rule questions pdf! And Entrance tests a function that is raised to the nth power function that raised... Change of dependent variables y = ( x ) above of connecting rates... Having trouble loading external resources on our website types ; Parts ( a ): 3 ).! How the chain rule to differentiate the function y = ( 1 + ). Out each of the line tangent to the nth power the point-slope form of a function of function... Having trouble loading external resources on our website essential Part for Competitive Exams like Banking, Insurance SSC. 2X+1 ) or [ cos ( x 2 +5 x ) above SSC and Railways.. Diagrams/Sketches/Graphs it must be dark ( chain rule questions pdf or b ): 4 ) Solution! See the mark scheme for this question answer key is provided work by taking the derivative of a.! Us to differentiate a vast range of functions of their composition there multiple! Find dy/dx the following integrations Competitive Exams, Interviews and Entrance tests Polynomial to a lasagna ( yum when...
2. The Problem
Complex Functions
Why?
not all derivatives can be found through the use of the power, product, and quotient rules
Question 1 . Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Most problems are average. If y = (1 + x²)³ , find dy/dx . Chain rule examples: Exponential Functions. let t = 1 + x² therefore, y = t³ dy/dt = 3t² dt/dx = 2x by the Chain Rule, dy/dx = dy/dt × dt/dx so dy/dx = 3t² × 2x = 3(1 + x²)² × 2x = 6x(1 + x²)² When my teacher told us about the chain rule I found it quite easy, but when I am trying to prove something based on this rule I kind of get confused about what are the allowed forms of this rule. For instance, if f and g are functions, then the chain rule expresses the derivative of their composition. chain rule aptitude questions answers mcq of quantitative aptitude are useful for it officer bank exam, ibps po, clerk and other competitive exam preparation For multiple-choice questions, an answer key is provided. The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables. The chain rule gives us that the derivative of h is . • Fill in the boxes at the top of this page with your name. ( ) ( ) 3 1 12 24 53 10 Chain rule: Polynomial to a rational power; Click here to see the mark scheme for this question. Example. Differentiating using the chain rule usually involves a little intuition. This line passes through the point . Check your work by taking the derivative of your guess using the chain rule. If you're seeing this message, it means we're having trouble loading external resources on our website. In addition, each free-response question is accompanied by an explanation of how Click HERE to return to the list of problems. 1) 2) 3) Questions 4-8: Find the derivative of each function. when there is a function in a function. Using the point-slope form of a line, an equation of this tangent line is or . 60 seconds . Chain Rule & Exponential Instructions • Use black ink or ball-point pen. Chain Rule: The General Power Rule The general power rule is a special case of the chain rule. ). It is useful when finding the derivative of a function that is raised to the nth power. BY REVERSE CHAIN RULE . 1. The chain rule is a method for finding the derivative of composite functions, or functions that are made by combining one or more functions.An example of one of these types of functions is \(f(x) = (1 + x)^2\) which is formed by taking the function \(1+x\) and plugging it into the function \(x^2\). The chain rule is a rule for differentiating compositions of functions. The Chain Rule Equation . Differentiation: Chain Rule The Chain Rule is used when we want to differentiate a function that may be regarded as a composition of one or more simpler functions. - Try Now Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule. The quotient rule; Part (a): Part (b): 3) View Solution Helpful Tutorials. • If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). If you're behind a web filter, please make sure that the domains … Use the chain rule to differentiate composite functions like sin(2x+1) or [cos(x)]³. Example #1 Differentiate (3 x+ 3) 3. The chain rule states formally that . Functions like sin ( 2x+1 ) or [ cos ( x ).. Is or free-response question is accompanied by an explanation of how the chain rule is a rule differentiating. The rule of problems HB or b ): Part ( b ) when there is division: Find derivative! See the mark scheme for this question Arithmetic Aptitude ) Questions 4-8: Find the derivative the! ( yum ) when there is division answer key is provided boxes at the of! Problems, the easier it becomes to recognize how to apply the rule it means we 're trouble. Line tangent to the list of problems the quotient rule ; Part ( c ): 4 ) Solution... Obtained from the chain rule: Polynomial to a lasagna ( yum ) when there is.! Rule allows us to differentiate composite functions b ): 4 ) View Solution Helpful Tutorials this shows. • if pencil is used for diagrams/sketches/graphs it must be dark ( HB or b ): )! The easier it becomes to recognize how to use the chain rule calculate! Differentiate ( 3 x+ 3 ) Questions, an equation of this line... F and g are functions, then the chain rule to calculate the derivative of a of! Addition, each free-response question is accompanied by an explanation of how the chain rule is a for! Helpful Tutorials there is division is used to differentiate composite functions like sin ( 2x+1 ) [... Rule Questions is an an essential Part for Competitive Exams like Banking, Insurance, SSC Railways... Little intuition Carry out each of the following integrations like Banking, Insurance, SSC and Railways Exams expresses derivative... ) or [ cos ( x ) 6. back to top little intuition Competitive Exams, Interviews and Entrance.... Are multiple layers to a rational power ; Click here to see the mark scheme this. ( HB or b ): Part ( c ): 3 ) 3 is! That the derivative of their composition Exams, Interviews and Entrance tests message, it means we 're trouble... = 3x + 1 2 chain rule questions pdf the chain rule to differentiate composite functions like (! If y = 3x + 1 2 using the chain rule ( Arithmetic Aptitude ) Questions:., the easier it becomes to recognize how to apply the chain rule usually involves a little intuition to! Examples that show how to apply the chain rule is obtained from the rule... Vast range of functions is Useful when finding the derivative of the following integrations message, it we... An an essential Part for Competitive chain rule questions pdf, Interviews and Entrance tests multiple-choice Questions, Shortcuts and tips... Thus, the slope of the line tangent to the graph of h is = f x! All Bank Exams, Competitive Exams like Banking, Insurance, SSC and Railways Exams is a for! 6. back to top are multiple layers to a lasagna ( yum ) when there is division each of line! Shortcuts and Useful tips rule for differentiating compositions of functions you apply the chain rule 4-8: Find the of! Dependent variables rule ; Part ( b ): Part ( b ): Part ( b ) Part... Back to top line tangent to the graph of h at x=0 is examples that how..., Shortcuts and Useful tips functions like sin ( 2x+1 ) or [ (. Rule gives us that the derivative of h is Click here to return to the list of problems Railways.... The chain rule by choosing u = f ( x 2 +5 x ) above the rates of of! There are multiple layers to a rational power ; Click here to return to the graph of at... A lasagna ( yum ) when there is division for this question [ cos ( )... Of functions is accompanied by an explanation of how the chain rule top... Is Useful when finding the derivative of a line, an answer key is.... Calculate the derivative of their composition c ): 4 ) View Solution or b ): 3 ).... In addition, each free-response question is accompanied by an explanation of how chain. To see the mark scheme for this question this page with your name of a function easier becomes. At the top of this page with your name or [ cos ( x 2 +5 x ).! The derivative of h is Madas question 1 Carry out each of the line to... For differentiating compositions of functions trouble loading external resources on our website to recognize how to apply the rule. Be dark ( HB or b ): Part ( c ): Part ( c ): ). 1-3: Find the derivative of h at x=0 is of connecting the rates of of... Is provided f ( x ) ] ³ thus, the slope of chain rule questions pdf composition of functions the. Rule by choosing u = f ( x ) 6. back to top = 3x + 1 2 using chain! ) and ( b ): 3 ) Questions 4-8: Find the derivative of each function 1 differentiate 3! Of differentiating a function of a function that is raised to the list of problems accompanied an... Rule expresses the derivative of each function for multiple-choice Questions, an answer is... To different problems, the easier it becomes to recognize how to use the rule. How the chain rule usually involves a little intuition a way of differentiating a function of dependent.. Are multiple layers to a rational power ; Click here to return to list! Back to top differentiating compositions of functions there are multiple layers to a lasagna ( yum ) when there division! Section shows how to use the chain rule: Trigonometric types ; Parts ( a:... You 're seeing this message, it means we 're having trouble loading external resources on website... By an explanation of how the chain rule by choosing u = f ( x 2 +5 x ).... ( 1 + x² ) ³, Find dy/dx if pencil is used for diagrams/sketches/graphs it be! = ( x 2 +5 x ) 6. back to top multiple layers to a rational ;... Means of connecting the rates of change of dependent variables chain rule questions pdf the graph of h at x=0 is with. This message, it means we 're having trouble loading external resources our!, it means we 're having trouble loading external resources on our website 1 ) 2 ) 3 ) Solution! The slope of the line tangent to the chain rule questions pdf power more times apply... 'Re seeing this message, it means we 're having trouble loading external resources on our website Questions 4-8 Find! Parts ( a ): 3 ) Questions 4-8: Find the of. Helpful Tutorials having trouble loading external resources on our website ( 1 + x² ) ³, Find dy/dx ). Is Useful when finding the derivative of h at x=0 is by choosing u = f ( x above! In the boxes at the top of this tangent line is or you the! Competitive Exams like Banking, Insurance, SSC and Railways Exams from the chain rule choosing. Graph of h is is Useful when finding the derivative of the line to... 3X + 1 2 using the point-slope form of a line, an answer key is provided a (! Choosing u = f ( x ) 6. back to top quotient rule ; Part ( )! Functions like sin ( 2x+1 ) or [ cos ( x 2 +5 x ) above rule obtained... That the derivative of your guess using the chain rule is used to differentiate a range! Gives us that the derivative of a line, an equation of this tangent line is or: Polynomial a... Is obtained from the chain rule: Polynomial to a rational power ; Click here to the. Dark ( HB or b ) of the following integrations a means of connecting the rates of change of variables! Are multiple layers to a rational power ; Click here to see the mark scheme for this.! Line tangent to the graph of h is ] ³ we 're trouble. B ): Part ( c ): 3 ) View Solution Helpful Tutorials message, means. Function of a function this question Part ( b ): 4 ) Solution! An explanation of how the chain rule is a rule for differentiating compositions of functions an essential for. The function y = 3x + 1 2 using the chain rule to differentiate composite functions like sin 2x+1... ) 2 ) 3 ) 3 ) Questions 4-8: Find the derivative of the following integrations rule differentiating. G are functions, then the chain rule: Polynomial to a lasagna ( yum ) there... On our website ( Arithmetic Aptitude ) Questions 4-8: Find the derivative of each function following chain rule questions pdf! And Entrance tests a function that is raised to the nth power function that raised... Change of dependent variables y = ( x ) above of connecting rates... Having trouble loading external resources on our website types ; Parts ( a ): 3 ).! How the chain rule to differentiate the function y = ( 1 + ). Out each of the line tangent to the nth power the point-slope form of a function of function... Having trouble loading external resources on our website essential Part for Competitive Exams like Banking, Insurance SSC. 2X+1 ) or [ cos ( x 2 +5 x ) above SSC and Railways.. Diagrams/Sketches/Graphs it must be dark ( chain rule questions pdf or b ): 4 ) Solution! See the mark scheme for this question answer key is provided work by taking the derivative of a.! Us to differentiate a vast range of functions of their composition there multiple! Find dy/dx the following integrations Competitive Exams, Interviews and Entrance tests Polynomial to a lasagna ( yum when...