Section 3.3 - Chain Rule

3.3 Chain Rule

  • The chain rule for derivatives applies to composed functions.

    • Given two functions f(x) and g(x), the composition is written as f ◦ g, meaning f(g(x)).

    • g is the input for function f, often described as f 'eating' g.

The Chain Rule Formula

  • If f is differentiable at g(x) and g is differentiable at x, then:

    • F(x) = f(g(x)) is differentiable at x,

    • Derivative: F'(x) = f'(g(x)) · g'(x)

  • In Leibniz notation, where y = f(u) and u = g(x), it is expressed as:

    • dy/dx = (dy/du) · (du/dx)

Proof of the Chain Rule

  • The definition of the derivative is given by:

    • f'(x) = lim(h→0) [f(x + h) − f(x)] / h

  • To find the derivative of f(x) at a value c, rewrite as:

    • f'(c) = lim(x→c) [f(x) − f(c)] / [x − c]

  • For proof, divide and multiply by g(x) − g(c):

    • Let h(x) = f(g(x))

    • Show for x = c that h'(c) = f'(g(c)) · g'(c)

    • Formulate:

      • lim(x→c) [f(g(x)) − f(g(c))] / [x − c]

      • This becomes: lim(x→c) [f(g(x)) − f(g(c))] / [g(x) − g(c)] · lim(x→c) [g(x) − g(c)] / [x − c]

      • Result: h'(c) = f'(g(c)) · g'(c)

  • As this holds for any c, it can simply be replaced by x.

Application of the Chain Rule

  • While it may seem complex, taking derivatives using the chain rule is similar to other derivatives, adding an extra step: finding the inside function.

Example 1: Finding f'(x)

  • Given f(x) = (6x² + 4x)³:

    • Inside: u = 6x² + 4x

    • Outside: y = u³

    • Derivatives: y' = 3u², u' = 12x + 4

    • Using chain rule: f'(x) = y' · u'

      • = 3u² · (12x + 4)

      • Replace u: f'(x) = 3(6x² + 4x)²(12x + 4)

    • Avoid complete simplification to prevent polynomial multiplication.

Additional Examples

  • Find the following derivatives using the chain rule (do not simplify):

    • (a) f(x) = (4x² + 3x − 9)⁵

    • (b) f(x) = (7x⁸ − 4x⁶)³

    • (c) f(x) = (3x⁴ + 5)⁶

Example 2: Differentiating f(x) = √(x³ − 9)

  • Inside: u = x³ − 9

  • Outside: y = √u = u^(1/2)

  • Derivatives: y' = (1/2)u^(−1/2) = (1/2√u), u' = 3x²

  • Using the chain rule: f'(x) = y' · u'

    • = (1/2√u)(3x²)

    • Replace u: f'(x) = (3x²)/(2√(x³ − 9))

More Practice Examples

  • Find the following derivatives using the chain rule (do not simplify):

    • (a) f(x) = √(x³ − 7x)

    • (b) f(x) = √(7 − 2x² + x − 3)

    • (c) f(x) = (x² − 1)^(2/3)

Chain Rule with Exponential and Logarithmic Functions

  • The chain rule is critical for taking derivatives of exponential and logarithmic functions.

  • Upcoming examples will further illustrate its application.

  • Practice finding the following derivatives using the chain rule (do not simplify):

    • (a) f(x) = −5e^(3x)

    • (b) f(x) = ln(x³ − 6x³)

    • (c) f(x) = (1/4)e^(4x)

    • (d) f(x) = 57x

    • (e) f(x) = log₇(5x)