chain rule
(noun)
a formula for computing the derivative of the composition of two or more functions.
Examples of chain rule in the following topics:
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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 $f \circ g(x) = f[g(x)]$Â 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
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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.
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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.
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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.
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The Natural Exponential Function: Differentiation and Integration
- Thus, we can use the limit rules to move it to the outside, leaving us with
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Derivatives of Exponential Functions
- Furthermore, for any differentiable function $f(x)$, we find, by the chain rule:
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Derivatives of Logarithmic Functions
- Applying the chain rule and the property of exponents we derived earlier, we can take the derivative of both sides:
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The Natural Logarithmic Function: Differentiation and Integration
- This is the case because of the chain rule and the following fact:
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Directional Derivatives and the Gradient Vector
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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