Chain Rule

Definition and Statement

Composite Functions

Composite function: function formed by applying one function to the result of another. Notation: if y = f(u) and u = g(x), then y = f(g(x)).

Chain Rule Statement

Derivative of composite function: product of derivative of outer function evaluated at inner function and derivative of inner function.

If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x)

Conditions for Use

Both f and g must be differentiable at relevant points. Applies to real-valued functions of one variable and extends to multivariable contexts.

Intuition and Geometric Interpretation

Rate of Change Perspective

Change in output y relative to x is product of change in y relative to u and change in u relative to x.

Visualizing Composition

Function g transforms domain, function f transforms range of g. Chain rule computes slope along combined transformation.

Geometric Analogy

Composite slope = slope of outer curve at inner function times slope of inner curve at point. Analogy: speed = rate of distance change × rate of time change.

Formal Proof

Using Limit Definition

Start with definition of derivative as limit of difference quotient. Express change in y as difference of f(g(x+h)) - f(g(x)).

Intermediate Variable Technique

Rewrite difference quotient to factor in change in g(x), then apply limits separately using differentiability of f and g.

Summary of Proof Steps

1. Write difference quotient for composite function.
2. Introduce intermediate increment.
3. Separate limits and apply continuity.
4. Conclude product of derivatives.

Basic Examples

Example 1: Polynomial Composition

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

Let u = 3x + 2dy/dx = 5u^4 * 3 = 15(3x + 2)^4

Example 2: Trigonometric Composition

Function: y = sin(x^2)

Let u = x^2dy/dx = cos(u) * 2x = 2x cos(x^2)

Example 3: Exponential Composition

Function: y = e^{3x^2 + 1}

Let u = 3x^2 + 1dy/dx = e^u * 6x = 6x e^{3x^2 + 1}

Applications in Calculus

Differentiating Complex Functions

Enables derivative calculation for nested functions, essential in real-world modeling and physics equations.

Related Rates Problems

Chain rule links rates of change of related variables over time. Crucial in mechanics, engineering, and biology.

Optimization and Curve Sketching

Used to find critical points of composite functions, infer concavity, and analyze behavior of curves.

Implicit Differentiation

Foundation for differentiating implicit functions where variables depend on each other.

Higher-Order Derivatives

Repeated Application

Second and higher derivatives computed by repeated use of chain and product rules.

Faà di Bruno's Formula

General formula for nth derivative of composite functions, extends chain rule systematically.

Example: Second Derivative

Given y = f(g(x)),dy/dx = f'(g(x)) * g'(x)d²y/dx² = f''(g(x)) * (g'(x))² + f'(g(x)) * g''(x)

Chain Rule in Multivariable Calculus

Functions of Several Variables

When functions depend on multiple variables, chain rule extends via partial derivatives.

General Formula

If z = f(x,y), and x = g(t), y = h(t), thendz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)

Jacobian Matrix Interpretation

Chain rule corresponds to matrix multiplication of Jacobians representing derivative mappings between coordinate systems.

Implicit Differentiation

Definition

Technique to differentiate equations defining variables implicitly rather than explicitly.

Use of Chain Rule

Apply chain rule to terms involving dependent variables inside functions.

Example

Given x² + y² = 25,differentiate both sides w.r.t x:2x + 2y (dy/dx) = 0dy/dx = -x/y

Common Mistakes

Forgetting to Differentiate Inner Function

Neglecting g'(x) results in incorrect derivatives.

Misidentifying Outer and Inner Functions

Incorrect function decomposition leads to errors in applying chain rule.

Ignoring Domain Restrictions

Failing to verify differentiability or domain may invalidate chain rule application.

Practice Problems

Problem 1

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

Problem 2

Find dy/dx if y = ln(sin x).

Problem 3

Compute derivative of y = sqrt(1 + e^{2x}).

Problem 4

Find dy/dt for z = x^2 y where x = t^2, y = sin t.

Problem 5

Implicitly differentiate xy + y^3 = 7 and find dy/dx.

Summary

Chain rule: essential calculus tool for differentiating compositions. Key formula: (f ∘ g)' = (f' ∘ g) · g'. Applies in single and multivariable contexts. Enables solving complex differentiation, related rates, implicit differentiation, and higher-order derivatives. Mastery crucial for advanced calculus and applications.

References

• Stewart, J. Calculus: Early Transcendentals. Brooks Cole, Vol. 8, 2015, pp. 150-170.
• Spivak, M. Calculus. Publish or Perish, Vol. 3, 2008, pp. 120-142.
• Thomas, G.B., Weir, M.D. Thomas' Calculus. Pearson, Vol. 14, 2017, pp. 220-245.
• Apostol, T.M. Calculus, Vol. 1: One-Variable Calculus. Wiley, 2007, pp. 98-115.
• Rudin, W. Principles of Mathematical Analysis. McGraw-Hill, 1976, pp. 180-200.