Chain Rule for differentiation

Chain Rule:

Rule:

  • For a composite function f(g(x)) where f(u) and g(x) are differentiable functions, the chain rule states: ddx[f(g(x))]=f(g(x))g(x).

Explanation:

  • The chain rule allows us to find the derivative of a function within a function. It considers the rate of change at one function nested within another and combines these rates to find the overall rate of change.

Example:

  • Consider f(x)=sin(x2)
    • Let u(x)=sin(x) and v(x)=x2
    • u(x)=cos(x) and v(x)=2x
    • Apply the chain rule: ddx[sin(x2)]=cos(x2)2x

Properties and Notes:

  1. Nested Functions:

    • The chain rule is applied when functions are nested within each other.
  2. Indirect and Direct Compositions:

    • It works for both direct compositions (f(g(x))) and indirect compositions (f(g(h(x)))).
  3. Multiple Functions in Sequence:

    • The chain rule extends to more complex functions where multiple functions are nested within each other.

Applications of the Chain Rule:

  • Trigonometric Functions:

    • Used in differentiation of compositions involving trigonometric functions, such as sin(x2) or cos(3x).
  • Logarithmic and Exponential Functions:

    • Applied in finding derivatives of compositions involving logarithmic and exponential functions, like e2x2 or ln(x3).