Chain Rule

A Chain Rule is a rule for differentiating compositions of a functions.

• Context:
  • It can be used for Derivative Computation.
  • It can be defined as Let [math]\displaystyle{ z=f(x,y) }[/math] be a function of two independent variables [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] where [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] are themselves functions of some independent variable [math]\displaystyle{ t }[/math]. Then the derivative of [math]\displaystyle{ z }[/math] with respect to [math]\displaystyle{ t }[/math] can be found out as [math]\displaystyle{ \frac{dz}{dt}=\frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt} }[/math]
• Example(s):
  • [math]\displaystyle{ z(x,y)=xy, \frac{dz}{dt}=\frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}=\frac{dz}{dt}= (y) (2t) + (x) (2)=(2t) (2t) + (t^2) (2)=6t^2 }[/math]
• See: Functional Composition, Pointwise Product, Multiplication Rule of Probability, Dependent Variable, Integral, Substitution Rule.

---

---

References

2014

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/chain_rule Retrieved:2014-8-24.
  • In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f and g are functions, then the chain rule expresses the derivative of their composition (the function which maps to ) in terms of the derivatives of f and g and the product of functions · as follows:  :[math]\displaystyle{ (f\circ g)'=(f'\circ g)\cdot g'. }[/math]

    The chain rule can also be written with a different notation for function composition. The meaning is identical.  :[math]\displaystyle{ f(g(x))' = f'(g(x)) g'(x) }[/math]

    If z is a function of a variable y, which is itself a function of x (see dependent variable), then z is also a function of x and the chain rule may be written  :[math]\displaystyle{ \frac {dz}{dx} = \frac {dz}{dy} \cdot \frac {dy}{dx} }[/math]

    In integration, the counterpart to the chain rule is the substitution rule.

---