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Chain Rule Formula

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Mastering the Chain Rule in Differentiation: Formulas, Examples, and Theorems

Comprehensive Definition, Description, Examples & Rules 

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Introduction

The chain rule is a fundamental concept in differe­ntial calculus that allows us to find the derivative of composite­ functions. It provides a formula for calculating how the function’s rate of change­ is related to the rate of change of its parts at a given point. The­ chain rule plays a crucial role in solving problems involving composite­ functions in calculus. 

Importance in Calculus

The primary significance of chain rule in calculus is:

  • It makes it very simple to differentiate the composite functions and find the derivative of these functions. 
  • You can find the composite functions of more than two functions, a notable feature of the chain rule. 
  • You can easily differentiate all the derivatives, and applying them makes your complex calculations very easy. 

The primary role of the chain rule is to solve complex equations and problems, which will help you determine the composite functions of these problems, making your calculations simple and easy. The chain rule has an effective role in determining the difficulties and differentiating these equations properly. 

Understanding the Chain Rule in Differentiation

The chain rule in differentiation means differentiating the terms one by one in the form of a chain. The differentiation starts from the outermost function and is then closely continuous to the innermost function of the equation. The chain rule of differentiation is how you can find the composite function of an equation. The rule expresses the derivative of the composite functions of the equation. 

It is essential to calculate the function’s derivative while using the chain rule, as in calculus, the derivatives are very important because they allow people to study how the function changes over time. The derivatives directly provide proper information about the direction of a function and how it moves at a particular point.

Significance in Various Fields

The significance of using the chain rule in different fields is:

  • Physics: You can primarily use the chain rule while doing physical calculations, especially in the various theories of Newton, you can see the use of the chain rule to calculate the function’s derivative and determine its answer.
  • Engineering: Mathematics and Engineering are correlated, and both have a lot of significance in human life. Many Civil engineers use the chain rule to solve effective calculations, making the calculation easy. 
  • Economics: The use of the chain rule is also very effective in macroeconomic calculations, and there are many areas where you need to use the chain rule to solve macroeconomic problems.

The Chain Rule Formula

The formal mathematical expression in which way you can denote the chain rule is:

  • dy/dx = dy/du du/dx

The proper notation of the formula can be done as:

dy/dx is the derivative of y concerning x

dy/du is derivative of y concerning u

du/dx is the derivative of u concerning x

The step-by-step explanation of how you can easily apply the chain rule formula in different types of composite functions are:

  • Identify the chain rule, which must be a composite function if you have to use the chain rule for the formula. 
  • Determine the outermost function and the innermost function for calculation. 
  • You need to determine the derivative of the outer function and then the derivative of the inner function. 
  • Multiply both the derivatives, then simplify the chain rule derivative easily. 

Chain Rule Examples

The series of chain rule examples which will help you to get a proper idea of the chain rule is:

Example1: Differentiate y = cos x 2 by using the chain rule 

Let u = x 2 so that y = cos u. 

 du/ dx = 2x 

dy/du = − sin u 

The chain rule says dy/dx = dy/du × du/dx

 and so dy/dx = − sin u × 2x 

−2x sin x²

Example 2: A is moving along the curve with the equation y = √(x3 + 56). When A is at (2,8), y is increasing at the rate of 2 units per second. How fast is x changing? Use the chain rule to determine your answer

To find: dx/dt

√(x3 + 56) and dy/dt = 2  / sec

dy/dx = (1/2)(x3 + 56)-1/2 (3×2)

=[(3/2) x2 ] / (x3 + 56)½

Applying the chain rule, dx/dt = dx/dy × dy/dt

Given dy/dx at x = 2

dy/dx at x = 2is [3(4)]/2√64

dy/dx =¾

dx/dy = 4/3

Thus dx/dt = 4/3 × 2 = 8/3

So,  x is changing at 8/3 units per second.

Example 3: Find the derivative of y= ln √x using the chain rule.

y = ln √x.

f(x) = y is a can be differentiated by the chain rule.

Assume that u = √x. Then y = ln u.

By the chain rule formula,

dy/dx = dy/du · du/dx

dy/dx = d/du (ln u) · d/dx (√x)

dy/dx = (1/u) · (1/(2√x))

dy/dx = (1/√x). (1/(2√x))

dy/dx = 1/(2x) (because u = 1/(2√x)).

y = cos (2×2 + 1).

So, dy/dx = 1/(2x)

Chain Rule Derivative

The chain rule derivative is the derivative of a composite function of the equation that helps to calculate the rate of change of the equation and is also one of the essential parts of the chain rule formula.

In differential calculus, the chain rule derivative is essential for calculating the rate of change of function if y = f{g(x)} f is relative to g. In contrast, g is relative to x, so the instantaneous rate suggests that f is relative to x. 

The real-world applications where understanding the chain rule derivative is important:

  • While changing the temperature variation, using the chain rule derivative is very important. 
  • While calculating the profit and loss in business using graphs, you can also use the chain rule derivative formula. 
  • We use the chain rule derivatives to calculate the speed and distance covered. 

Using the Chain Rule for Composite Functions

The process of using the chain rule for composite functions is that the chain rule is possible only for the composite function, as the formula is not possible for any other function. The outside and inside rule is only possible for the composite function, and so where the derivative occurs first on the outermost function and then on the innermost function. 

Challenges and Pitfalls

The common challenges and pitfalls while using the chain rule formula are:

  • Not recognizing whether the function is a composite function or a different function.
  • Forgetting to multiply the derivative by the innermost function and only using the outermost function. 
  • Failing to identify the proper outermost or the innermost function. 
  • Computing the formula in the wrong way. 

Tips and Strategies

The tips and strategies that you can follow to avoid any challenges in the chain rule formula are:

  • Identify your function properly and use the chain rule formula only for composite functions. 
  • Identify the inner function and the outer function properly. 
  • Find the derivative of the outer function first and then move on to the inner function, after which you can multiply both of them. 

The Chain Rule Theorem

The chain rule theorem is a particular theorem of the composite functions that states the derivative of that particular composite function has a direct relation to the chain rule formula and extends the applicability of the chain rule. 

Many theories are related to the chain rule formula, especially in engineering and mathematics. The theory suggests that for the computation of the derivative of the chain rule theorem, it is sufficient for you to compute the derivative of the outermost and the innermost functions. According to a theorem, the formula will be perfect for calculating the hardest equations.

How the theorem enhances the applicability of the chain rule in various scenarios is that the composite function:

  • f{g(x)} = f{g(x)} × g(x)

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Key Takeaways

  1. The chain rule is a primary concept that makes the calculation easy, and you can make complex calculations very simple. 

  2. The outermost and the innermost functions are the most important to use the chain rule. 

  3. The derivative of the outermost function happens first, following which the innermost function is used.

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Frequently Asked Questions

If f(x)=6x+3 and g(x)=−2x+5 Use the chain rule to calculate h′(x)

, where h(x)=f(g(x))

The derivatives of and g are:

f′(x))= 6

g′(x= −2.

According to the chain rule,

h′(x)=f′(g(x))g′(x)

=f′(−2x+5)(−2)

=6(−2)

=−12.

So, h′(x)=−12

The chain rule derivative represents the composite functions that determine both differentiable functions and define these functions to evaluate the actual equation and simplify the equation.

Yes, the chain rule can have different variations, and you can use the chain rule to define it in two forms. If the f and g functions have two differentiable functions, then you can have different variations for the chain rule.

Yes, there are a few common mistakes to avoid while using the chain rule, and these mistakes are:

  • Not recognizing if the function you are following is a composite or a different function.
  • Not identifying the proper outermost or the innermost function and failing to calculate the formula. 
  • Not finding out the derivative of the inner function and only using the outer function for calculation. 
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2D Shapes2cosacosb Formula30-60-90 Formulas3D ShapesAbsolute Value FormulaAcute AngleAcute Angle triangleAdditionAlgebra FormulasAlgebra of MatricesAlgebraic EquationsAlgebraic ExpressionsAngle FormulaAnnulusAnova FormulaAnti-derivative FormulaAntiderivative FormulaApplication of DerivativesApplications of IntegrationArc Length FormulaArccot FormulaArctan FormulaArea Formula for QuadrilateralsArea FormulasArea Of A Sector Of A Circle FormulaArea Of An Octagon FormulaArea Of Isosceles TriangleArea Of ShapesArea Under the Curve FormulaArea of RectangleArea of Regular Polygon FormulaArea of TriangleArea of a Circle FormulaArea of a Pentagon FormulaArea of a Square FormulaArea of a Trapezoid FormulaArithmetic Mean FormulaArithmetic ProgressionsArithmetic Sequence Recursive FormulaArithmetic and Geometric ProgressionAscending OrderAssociative Property FormulaAsymptote FormulaAverage Deviation FormulaAverage Rate of Change FormulaAveragesAxioms Of ProbabilityAxis of Symmetry FormulaBasic Math FormulasBasics Of AlgebraBinary FormulaBinomial Probability FormulaBinomial Theorem FormulaBinomial distributionBodmas RuleBoolean AlgebraBusiness MathematicsCalculusCelsius FormulaCentral Angle of a Circle FormulaCentral Limit Theorem FormulaCentroid of a Trapezoid FormulaChain RuleChain Rule FormulaChange of Base FormulaChi Square FormulaCirclesCircumference FormulaCoefficient of Determination FormulaCoefficient of Variation FormulaCofactor FormulaComplete the square formulaComplex numbersCompound Interest FormulaConditional Probability FormulaConeConfidence Interval FormulaCongruence of TrianglesCorrelation Coefficient FormulaCos Double Angle FormulaCos Square theta FormulaCos Theta FormulaCosec Cot FormulaCosecant FormulaCosine FormulaCovariance FormulaCubeCurated Maths Resources for Teachers – EdulyteCylinderDecimalsDifferential calculusDiscover the world of MathsEllipseEquilateral triangleEuler’s formulaEven numbersExponentsFractionFraction to decimalGeometric sequenceHeptagonHyperbolaIntegersIntegrationIntegration by partsLinesLocusMatricesNatural numbersNumber lineOdd numbersParallelogramPercentage formulaPerimeterPolygonPolynomialsPrismProbabilityPyramidPythagoras theoremRoman NumeralsScalene triangleSetsShapes NamesSimple interest formulaSlope formulaSolid shapesSphereSquareStandard deviation formulaSubtractionSymmetryTimeTrianglesTrigonometry formulaTypes of anglesValue of PiVariance formulaVectorVolume formulasVolume of a coneVolume of sphere formulaWhole numbers
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