Comprehensive Study Notes on the Chain Rule and Function Composition

Foundations of Function Composition and the Chain Rule

• Existing Derivative Rules:

  • Sum and Difference Rule (addition and subtraction of functions).

  • Product Rule (multiplication of functions).

  • Quotient Rule (division of functions).

  • Constant Rule and Constant Multiple Rule.

• Function Composition:

  • Composition is an additional method for combining two functions, denoted as $h(x) = f(g(x))$

  • Composition occurs when the range of an inner function $g(x)$ matches the domain of an outer function $f(x)$.

  • The inner function $g(x)$ is substituted directly in place of the independent variable in $f(x)$.

Mathematical Derivation of the Chain Rule

• Definition of Derivative at a Point:

  • To find the derivative of $h(x) = f(g(x))$ at a specific point $a$, start with the limit definition of the derivative:         $h'(a) = \lim_{x \to a} \frac{h(x) - h(a)}{x - a}$

  • Substituting $f(g(x))$ for $h(x)$ and $f(g(a))$ for $h(a)$:         $h'(a) = \lim_{x \to a} \frac{f(g(x)) - f(g(a))}{x - a}$

• Algebraic Manipulation Using Identity Multiplication:

  • Multiply the expression by a fraction equivalent to $1$:         $\frac{g(x) - g(a)}{g(x) - g(a)}$

  • This yields the expanded limit expression:         $h'(a) = \lim_{x \to a} \left( \frac{f(g(x)) - f(g(a))}{x - a} \times \frac{g(x) - g(a)}{g(x) - g(a)} \right)$

  • Swap the denominators of the two fractions:         $h'(a) = \lim_{x \to a} \left( \frac{f(g(x)) - f(g(a))}{g(x) - g(a)} \times \frac{g(x) - g(a)}{x - a} \right)$

• Applying Limit Laws:

  • The limit of a product is equal to the product of the individual limits, provided both limits exist:         $h'(a) = \left( \lim_{x \to a} \frac{f(g(x)) - f(g(a))}{g(x) - g(a)} \right) \times \left( \lim_{x \to a} \frac{g(x) - g(a)}{x - a} \right)$

• Evaluating the Individual Limits:

  • The second factor is recognized directly as the derivative of $g(x)$ at point $a$:         $\lim_{x \to a} \frac{g(x) - g(a)}{x - a} = g'(a)$

  • The first factor represents the slope of the secant line passing through the points $(g(x), f(g(x)))$ and $(g(a), f(g(a)))$.

  • As $x \to a$, the term $g(x)$ approaches $g(a)$, making this limit evaluate to the derivative of $f(x)$ at the point $g(a)$:         $\lim_{x \to a} \frac{f(g(x)) - f(g(a))}{g(x) - g(a)} = f'(g(a))$

  • Combining these results yields the derivative at point $a$:         $h'(a) = f'(g(a)) \times g'(a)$

• Generalization to All Points:

  • While $a$ denotes a specific point, this relation holds for all arbitrary points $x$ in the domain:         $h'(x) = f'(g(x)) \times g'(x)$

Verbal Definition and Tabular Framework

• Verbal Statement:

  • The derivative of a composite function is the derivative of the outside function evaluated at the inside function, multiplied by the derivative of the inside function.

• Symbolic Component Breakdown:

  • Derivative of outside function evaluated at inside function: $f'(g(x))$

  • Derivative of inside function: $g'(x)$

• Tabular Method for Derivative Evaluation:

  • Identify the outside function $f(x)$.

  • Identify the inside function $g(x)$.

  • Compute the derivative of the outside function $f'(x)$.

  • Evaluate the derivative of the outside function at the inside function $f'(g(x))$.

  • Compute the derivative of the inside function $g'(x)$.

  • Multiply $f'(g(x))$ by $g'(x)$ to form $\frac{dy}{dx}$.

Applied Examples and Step-by-Step Solutions

Example 1: Composite Power of a Trigonometric Function

• Function:     $y = (\sin(x))^2$

• Component Identification:

  • Outside function: $f(x) = x^2$

  • Inside function: $g(x) = \sin(x)$

• Derivatives Computation:

  • Derivative of outside function via Power Rule (bring power to front, subtract 1 from power):         $f'(x) = 2x^1 = 2x$

  • Outside derivative evaluated at inside function:         $f'(g(x)) = 2\sin(x)$

  • Derivative of inside function:         $g'(x) = \cos(x)$

• Final Differentiation:

  • Multiply components:         $\frac{dy}{dx} = f'(g(x)) \times g'(x) = 2\sin(x)\cos(x)$

  • Consistency Check: Evaluating $y = (\sin(x))^2 = \sin(x)\sin(x)$ via the Product Rule yields $(\sin(x))'\sin(x) + \sin(x)(\sin(x))' = \cos(x)\sin(x) + \sin(x)\cos(x) = 2\sin(x)\cos(x)$, confirming mathematical validity.

Example 2: Radical Function with Linear Argument

• Function:     $y = \sqrt{x + 1}$

• Component Identification:

  • Outside function: $f(x) = \sqrt{x} = x^{\frac{1}{2}}$

  • Inside function: $g(x) = x + 1$

• Derivatives Computation:

  • Derivative of outside function via Power Rule:         $f'(x) = \frac{1}{2}x^{\frac{1}{2} - 1} = \frac{1}{2}x^{-\frac{1}{2}}$

  • Outside derivative evaluated at inside function:         $f'(g(x)) = \frac{1}{2}(x + 1)^{-\frac{1}{2}} = \frac{1}{2\sqrt{x + 1}}$

  • Derivative of inside function (Sum Rule):

    • Power Rule on $x$ gives $1x^0 = 1$

    • Constant Rule on $1$ gives $0$

    • $g'(x) = 1 + 0 = 1$

• Final Differentiation:

  • Multiply components:         $\frac{dy}{dx} = f'(g(x)) \times g'(x) = \frac{1}{2\sqrt{x + 1}} \times 1 = \frac{1}{2\sqrt{x + 1}}$

Example 3: Trigonometric Function with Constant Multiple

• Function:     $y = \cos(3t)$

• Component Identification (using independent variable $t$):

  • Outside function: $f(t) = \cos(t)$

  • Inside function: $g(t) = 3t$

• Derivatives Computation:

  • Derivative of outside function:         $f'(t) = -\sin(t)$

  • Outside derivative evaluated at inside function:         $f'(g(t)) = -\sin(3t)$

  • Derivative of inside function via Constant Multiple Rule:         $g'(t) = 3$

• Final Differentiation:

  • Multiply components:         $\frac{dy}{dt} = f'(g(t)) \times g'(t) = -\sin(3t) \times 3 = -3\sin(3t)$

Example 4: Doubly Nested Composite Function

• Function:     $y = 19\sin^2(x^7) = 19(\sin(x^7))^2$

• Structural Analysis:

  • This function contains a triple composition (a function inside a function inside a function).

  • Differentiate by working systematically from the outside in.

  • The constant coefficient $19$ is factored out during differentiation.

• Outer Level Decomposition:

  • Outside function: $f(x) = x^2$

  • Inside function: $g(x) = \sin(x^7)$

  • Derivative of outer function: $f'(x) = 2x$

  • Outer derivative evaluated at inside: $f'(g(x)) = 2\sin(x^7)$

• Inner Level Decomposition (Chain Rule applied to $g(x)$):

  • Let $g(x) = h(k(x))$

  • Outer inner function: $h(x) = \sin(x)$

  • Inner inner function: $k(x) = x^7$

  • Derivative of $h(x)$: $h'(x) = \cos(x)$

  • $h'$ evaluated at $k(x)$: $h'(k(x)) = \cos(x^7)$

  • Derivative of $k(x)$ via Power Rule: $k'(x) = 7x^{7-1} = 7x^6$

  • Derivative of $g(x)$:         $g'(x) = h'(k(x)) \times k'(x) = 7x^6\cos(x^7)$

• Combining All Components:

  • Include initial coefficient $19$:         $\frac{dy}{dx} = 19 \times f'(g(x)) \times g'(x)$         $\frac{dy}{dx} = 19 \times 2\sin(x^7) \times 7x^6\cos(x^7)$

• Arithmetic Multiplication Breakdown:

  • Multiply scalar constants:         $19 \times 2 \times 7 = 19 \times 14$

  • Expanding $19 \times 14$:         $(10 + 9) \times 14 = 100 + 90 + 40 + 36 = 266$

• Final Simplified Derivative:     $\frac{dy}{dx} = 266x^6\sin(x^7)\cos(x^7)$     (where $x^7$ is the explicit argument of both the sine and cosine functions).