2.2.1: Resources and Key Concepts

Math 400: Calculus I - Differential Calculus · LibreTexts

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    From LibreTexts, under CC BY-SA, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Resources

    Videos

    Key Concepts

    Theorems

    • Theorem: The Product Rule: Let \(f(x)\) and \(g(x)\) be differentiable functions. Then \(\frac{d}{dx}(f(x)g(x)) = \left(\frac{d}{dx}(f(x))\right) \cdot g(x) + \left(\frac{d}{dx}(g(x))\right) \cdot f(x)\).
      • Alternatively, if \(p(x) = f(x)g(x)\), then \(p^{\prime}(x) = f^{\prime}(x)g(x) + g^{\prime}(x)f(x)\).
    • Theorem: The Quotient Rule: Let \(f(x)\) and \(g(x)\) be differentiable functions. Then \(\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{\left(\frac{d}{dx}(f(x))\right) \cdot g(x) - \left(\frac{d}{dx}(g(x))\right) \cdot f(x)}{(g(x))^2}\).
      • Alternatively, if \(q(x) = \frac{f(x)}{g(x)}\), then \(q^{\prime}(x) = \frac{f^{\prime}(x)g(x) - g^{\prime}(x)f(x)}{(g(x))^2}\).
    • (Proof of the Power Rule for negative integers is presented using the Quotient Rule).

    Common Mistakes

    • The Product Rule is NOT the Product of Derivatives!: \(\frac{d}{dx}(f(x) \cdot g(x)) \neq \left(\frac{d}{dx}(f(x))\right) \cdot \left(\frac{d}{dx}(g(x))\right)\).
    • The Quotient Rule is NOT the Quotient of Derivatives!: \(\frac{d}{dx}\left(\frac{N(x)}{D(x)}\right) \neq \frac{\frac{d}{dx}(N(x))}{\frac{d}{dx}(D(x))}\).

    This page titled 2.2.1: Resources and Key Concepts was last modified on Mon, 22 Sep 2025 20:41:53 GMT and is shared under a CC BY-SA license and was authored, remixed, and/or curated by Roy Simpson.