Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Calculus → Differentiation ↓


Chain Rule


The chain rule is a fundamental concept in calculus that allows us to find the derivative of mixed functions. A mixed function is a function that is composed of two or more simple functions. Understanding how to differentiate these types of functions is essential to solving more complex problems in calculus.

To understand the chain rule, let's start with the concept of a composite function. Suppose we have two functions ( f(x) ) and ( g(x) ). A composite function occurs when one function is applied to the result of another function, written as ( (f circ g)(x) = f(g(x)) ). The chain rule helps us differentiate such composite functions.

Chain rule formula

Mathematically, the chain rule is expressed as:

If ( y = f(g(x)) ), then ( frac{dy}{dx} = f'(g(x)) cdot g'(x) ).

This means that to find the derivative of ( y = f(g(x)) ), you need to take the derivative of the outer function ( f ) with respect to the inner function ( g(x) ), and then multiply it by the derivative of the inner function ( g(x) ) with respect to ( x ).

Understanding chain rule through examples

Let's illustrate the chain rule with specific examples. Suppose you have this function:

y = (3x^2 + 5)^4

This is a composite function where the inner function is ( u = 3x^2 + 5 ) and the outer function is ( y = u^4 ). To find the derivative ( frac{dy}{dx} ), we follow these steps:

  1. Differentiate the outer function with respect to ( u ):
    2.     ( frac{dy}{du} = 4u^3 )
  2. Differentiate the inner function with respect to ( x ):
    3.     ( frac{du}{dx} = 6x )
  3. Apply the chain rule:
    4.     ( frac{dy}{dx} = frac{dy}{du} cdot frac{du}{dx} = 4(3x^2 + 5)^3 cdot 6x )
    Multiply to simplify: 
        ( frac{dy}{dx} = 24x(3x^2 + 5)^3 )

The chain rule allowed us to find the derivative of a more complicated function by breaking it down into more manageable parts.

Visual example

Consider plotting the function ( f(g(x)) = (2x + 3)^3 ) step-by-step using the chain rule. This visual representation can provide more information about how the rule works.

( f(g(x)) )

The curve above shows what ( (2x + 3)^3 ) might look like when graphed. As you move along the x-axis, the change in values depends not only on ( x ) but also on the inner function, which involves combinations.

Why use the chain rule?

The chain rule is important because many functions in mathematics are inherently mixed. Mixed functions are often used in real-world scenarios and physics problems. Calculus, especially differentiation, relies on breaking down complex scenarios into derivatives to understand rates of change and motion.

More text examples

Example 1: Differentiate ( y = sin(5x^3) ).

Here, the outer function is ( sin(u) ) where ( u = 5x^3 ). Follow these steps:

  1. Differentiate the outer function: ( frac{dy}{du} = cos(u) )
  2. Differentiate the inner function: ( frac{du}{dx} = 15x^2 )
  3. Apply chain rule: ( frac{dy}{dx} = cos(5x^3) cdot 15x^2 )
  4. Result: ( frac{dy}{dx} = 15x^2 cos(5x^3) )

Example 2: Differentiate ( y = e^{2x^2 - 3x} ).

The outer function is ( e^u ) where ( u = 2x^2 - 3x ).

  1. Differentiate the outer function: ( frac{dy}{du} = e^u )
  2. Differentiate the inner function: ( frac{du}{dx} = 4x - 3 )
  3. Apply the chain rule: ( frac{dy}{dx} = e^{2x^2 - 3x} cdot (4x - 3) )
  4. Result: ( frac{dy}{dx} = (4x - 3)e^{2x^2 - 3x} )

Conceptual visualization

Breaking down the derivative of a function using the chain rule is like peeling an onion layer by layer, analyzing the transformations made by each function layer involved in creating the overall function. Consider each function as a transformation in itself, and understanding each transformation helps us analyze how the final transformation occurs within the entire function.

Layers of functions

In the above illustration, the innermost circle might represent the main simple function. Subsequent layers represent each function added on top of the previous function, creating more complexity and defining how each modification affects the results.

Conclusion

The chain rule is a powerful tool in calculus. It helps us find the derivative of a composite function by understanding the effect of each function layer. By using a systematic approach, breaking down the process into manageable parts, and practicing with a variety of examples, the chain rule becomes an indispensable part of any calculus toolkit.

As you continue to learn and practice calculus, you will encounter more complex tasks that require differentiation. The chain rule will play a key role in solving these problems, giving you the flexibility to apply calculus to many real-world scenarios. Considering these principles will deepen your understanding and allow you to approach calculus with more confidence.


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Chain Rule

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