2.4.1: Resources and Key Concepts
- Page ID
- 192945
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Key Concepts
Theorems
- Theorem: The Chain Rule (Lagrange's Notation): Let \(f\) and \(g\) be functions. For all \(x\) in the domain of \(g\) for which \(g\) is differentiable at \(x\) and \(f\) is differentiable at \(g(x)\), the derivative of the composite function \(h(x) = (f \circ g)(x) = f(g(x))\) is given by \(h^{\prime}(x) = f^{\prime}(g(x)) \cdot g^{\prime}(x)\).
- Theorem: Power Rule for Composition of Functions (General Power Rule): For all values of \(x\) for which the derivative is defined, if \(h(x) = (g(x))^n\), then \(h^{\prime}(x) = n(g(x))^{n-1} \cdot g^{\prime}(x)\).
- Corollary: Chain Rule for a Composition of Three Functions: If \(k(x) = h(f(g(x)))\), then \(k^{\prime}(x) = h^{\prime}(f(g(x))) \cdot f^{\prime}(g(x)) \cdot g^{\prime}(x)\), for all values of \(x\) for which the function is differentiable.
- Theorem: Chain Rule (Leibniz's Notation): If \(y\) is a function of \(u\), and \(u\) is a function of \(x\), then \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\).
Common Mistakes
- Forgetting the "Inner" Derivative: A common mistake when first learning the Chain Rule is to differentiate the "outer" function but forget to multiply by the derivative of the "inner" function. For example, incorrectly stating \(\frac{d}{dx}(\sin(x^3)) = \cos(x^3)\) instead of \(\cos(x^3) \cdot 3x^2\).
- Incorrect Application to Products/Quotients: Confusing the Chain Rule with the Product or Quotient Rule, or misapplying them in combination.
- Leibniz Notation Final Answer: When using the Leibniz form of the Chain Rule, the final answer must be expressed entirely in terms of the original independent variable (e.g., \(x\)), not intermediate variables (e.g., \(u\)).