Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Calculus → Differential Calculus ↓


Chain Rule


One of the most important concepts to understand in calculus, especially differential calculus, is the chain rule. The chain rule is an important differential rule that tells how to find the derivative of a composite function. Let us understand this concept in detail to understand its applications, relevance, and underlying intuition.

What is the chain rule?

At its core, the chain rule provides a method for finding the derivative of a composite function. A composite function is essentially a function that is made up of two or more functions. For example, if you have a function y = f(g(x)), then f(g(x)) is a composite function where g(x) is a function nested within f.

The chain rule states that if a function y = f(u) is differentiable at u = g(x), and u = g(x) is differentiable at x, then the composite function y = f(g(x)) is differentiable at x, and:

    dy/dx = (dy/du) * (du/dx)

This may seem abstract at first glance, so let’s understand it with examples and visual aids.

Visualizing the chain rule

Consider a composite function y = f(g(x)). The process of the chain rule can be visualized as follows:

X u=g(x) y=f(u) dy/dx

In the graph, imagine that x is changing slightly. This small change in x causes a change in u = g(x), and therefore another change in y = f(u). The chain rule summarizes these interdependent changes.

Example: Applying the chain rule

Let's apply the chain rule to a specific function. Suppose we have this:

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

We can define:

  • u = 3x^2 + 2x
  • Thus, y = u^5

Now, let's find each derivative:

  1. First, differentiate u = 3x^2 + 2x with respect to x:
                du/dx = 6x + 2
  2. Next, differentiate y = u^5 with respect to u:
                dy/du = 5u^4

Applying the chain rule:

    dy/dx = (dy/du) * (du/dx) = 5u^4 * (6x + 2) = 5(3x^2 + 2x)^4 * (6x + 2)

Why is the chain rule important?

The chain rule is very important in calculus because it allows us to handle complex derivatives that are otherwise difficult to break down. Whenever functions are nested, the chain rule is incredibly useful and becomes an essential tool in any mathematician or scientist's toolkit. It helps to understand and solve problems that involve rates of change where multiple factors are involved.

General formula for the chain rule

In a more abstract sense, if you have a composite function y = f(g(h(x))), then using the chain rule repeatedly, we have:

    dy/dx = (dy/du) * (du/dv) * (dv/dx) = (df/dg) * (dg/dh) * (dh/dx)

Here, each re-application of the rule allows you to tackle another layer of composition.

Example with multiple layers

Let's consider:

    y = (sin(x^2 + 1))^3

Let's define the inner layers:

  • v = x^2 + 1
  • u = sin(v)
  • Then, y = u^3

Now, let us understand each step separately:

  1. Differentiate y = u^3 with respect to u:
                dy/du = 3u^2
  2. Differentiate u = sin(v) with respect to v:
                du/dv = cos(v)
  3. Differentiate v = x^2 + 1 with respect to x:
                dv/dx = 2x

Combination using Chain Rule:

    dy/dx = (dy/du) * (du/dv) * (dv/dx) = 3u^2 * cos(v) * 2x = 3(sin(x^2 + 1))^2 * cos(x^2 + 1) * 2x

Practical applications

The utility of the chain rule extends far beyond simple mathematical exercises; it is widely used in real-world problems related to physics, engineering, economics, biological sciences, and more. For example, in physics, the chain rule is often used to relate rates of change in different coordinate systems or to handle transformations in space.

Consider an example in physics where we need to differentiate a function representing the position of a particle in space with respect to time, and where the position depends on other dynamical variables.

Suppose the position s of an object is a function of the velocity v(t), which itself depends on the acceleration a(t):

    s = f(v(t)) v = g(a(t))

Apply the chain rule to find ds/dt:

    ds/dt = (ds/dv) * (dv/da) * (da/dt)

In this way, complex and layered dependencies in dynamic systems can be efficiently analyzed and understood.

Conclusion

The chain rule is a powerful and versatile tool within calculus that simplifies the differentiation of composite functions, allowing for deeper exploration into the relationships between variables. Mastery of this rule opens up many opportunities in both theoretical and applied mathematics, providing clarity in tackling complex problems involving layered functional relationships.

Constant practice, use of visual aids, and solving various exercises will strengthen understanding and help in efficient application of the chain rule in different contexts.


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Chain Rule

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