Chain Rule in Calculus

The chain rule is essential for differentiating intricate functions where one function is nested inside another, referred to as composite functions. This guide aims to assist you in understanding and applying the chain rule proficiently.

Composite Functions

• Composite Functions: Merge two or more functions where the output of one function is utilised as the input for another.
  • Notation: $f(g(x))$ visually represents this composition.
  • Example: Given $f(x) = x^2$ and $g(x) = 3x + 1$, then $f(g(x)) = (3x + 1)^2$.

The Chain Rule

• Mathematical Notation:

  • If $h(x) = f(g(x))$, then $h'(x) = f'(g(x)) \cdot g'(x)$.
  • Alternative Notation: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

• Purpose: This rule is beneficial for differentiating functions nested within other functions, permitting differentiation of composite functions by considering each function individually.

Worked Examples

Example 1: Differentiating $(3x + 1)^2$

• Identify Functions:

  • Outer: $f(u) = u^2$
  • Inner: $g(x) = 3x + 1$

• Differentiate:

  • $f'(u) = 2u$
  • $g'(x) = 3$

• Apply Chain Rule: $h'(x) = 2(3x + 1) \cdot 3 = 18x + 6$

Example 2: Differentiating $(3x^2 + 2)^5$

• Identify Functions:

  • Outer: $f(u) = u^5$
  • Inner: $g(x) = 3x^2 + 2$

• Differentiate:

  • $f'(u) = 5u^4$
  • $g'(x) = 6x$

• Apply Chain Rule: $\frac{dh}{dx} = 5(3x^2 + 2)^4 \cdot 6x = 30x(3x^2 + 2)^4$

Common Mistakes and Tips

Typical Errors

• Neglecting the Derivative of the Inner Function: Omitting multiplication by $g'(x)$ can significantly alter the outcome.
• Misidentifying Functions: Incorrectly determining which part is the inner versus the outer function can lead to errors.
• Misapplication of the Rule: Incorrect sequence of differentiation steps can produce erroneous results; ensure each derivative is applied accurately.

Strategies

• Use Visual Aids: Diagrams assist in clearly delineating the structure of functions.
• Mnemonic Devices: "Inside first, then out!" is helpful for remembering the sequence.
• Rechecking: A systematic review of each step ensures precision.

Practice Problems

Problems

1. Differentiate $f(x) = (x^2 + 1)^4$.
2. Differentiate $g(x) = \ln(5x^2 + 2)$.
3. Differentiate $h(x) = \sin(3x^2+ x)$.

Solutions

Exercise 1 Solution:

• Functions: $\text{outer } f(u) = u^4$, $\text{inner } g(x) = x^2 + 1$
• $\frac{df}{dx} = 4(x^2 + 1)^3 \cdot 2x = 8x(x^2 + 1)^3$

Exercise 2 Solution:

• Functions: $\text{outer } f(u) = \ln(u)$, $\text{inner } g(x) = 5x^2 + 2$
• $\frac{dg}{dx} = \frac{1}{5x^2 + 2} \cdot 10x = \frac{10x}{5x^2 + 2}$

Exercise 3 Solution:

• Functions: $\text{outer } f(u) = \sin(u)$, $\text{inner } g(x) = 3x^2 + x$
• $\frac{dh}{dx} = \cos(3x^2 + x) \cdot (6x + 1)$