chain rule

Examples of chain rule in the following topics:

• The Chain Rule

  • The chain rule is a formula for computing the derivative of the composition of two or more functions.
  • For example, following the chain rule for $f \circ g(x) = f[g(x)]$ yields:
  • Using the chain rule yields:
  • Use of the chain rule is needed for the complicated calculation.
  • Calculate the derivative of a composition of functions using the chain rule

• The Chain Rule

  • The chain rule is a formula for computing the derivative of the composition of two or more functions.
  • For example, the chain rule for $f \circ g$ is $\frac {df}{dx} = \frac {df}{dg} \, \frac {dg}{dx}$.
  • The chain rule above is for single variable functions $f(x)$ and $g(x)$.
  • However, the chain rule can be generalized to functions with multiple variables.
  • The chain rule can be used to take derivatives of multivariable functions with respect to a parameter.

• Differentiation Rules

  • The rules of differentiation can simplify derivatives by eliminating the need for complicated limit calculations.
  • In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided by using differentiation rules.
  • Some of the most basic rules are the following.
  • Here the second term was computed using the chain rule and the third using the product rule.
  • The flight of model rockets can be modeled using the product rule.

• Implicit Differentiation

  • Implicit differentiation makes use of the chain rule to differentiate implicitly defined functions.
  • Implicit differentiation makes use of the chain rule to differentiate implicitly defined functions.

• The Natural Exponential Function: Differentiation and Integration

  • Thus, we can use the limit rules to move it to the outside, leaving us with

• Derivatives of Exponential Functions

  • Furthermore, for any differentiable function $f(x)$, we find, by the chain rule:

• Derivatives of Logarithmic Functions

  • Applying the chain rule and the property of exponents we derived earlier, we can take the derivative of both sides:

• The Natural Logarithmic Function: Differentiation and Integration

  • This is the case because of the chain rule and the following fact:

• Directional Derivatives and the Gradient Vector

• The Substitution Rule

  • It is the counterpart to the chain rule of differentiation.
  • Use $u$-substitution (the substitution rule) to find the antiderivative of more complex functions