2.1.1: Resources and Key Concepts
- Page ID
- 192933
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Videos
- Derivative rules
- Constant Rule
- Power Rule
- Sum Rule for derivatives
- Derivatives of polynomials
- Tangent and normal lines
Key Concepts
Theorems
- Theorem: The Constant Rule: Let \(c\) be a constant. If \(f(x) = c\), then \(f'(x) = 0\). (Alternatively, \(\frac{d}{dx}(c) = 0\).)
- Theorem: The Power Rule (positive integer exponents only): Let \(n\) be a positive integer. If \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\). (Alternatively, \(\frac{d}{dx}(x^n) = nx^{n-1}\).)
- Theorem: The Power Rule (General Version): Let \(n \in \mathbb{R}\). If \(f(x) = x^n\), then \(\frac{d}{dx}(x^n) = nx^{n-1}\).
- Theorem: Sum, Difference, and Constant Multiple Rules: Let \(f(x)\) and \(g(x)\) be differentiable functions and \(k\) be a constant. Then, each of the following equations holds:
- Sum Rule: \(\frac{d}{dx}(f(x)+g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))\). (Alternatively, for \(s(x)=f(x)+g(x)\), \(s'(x)=f'(x)+g'(x)\).)
- Difference Rule: \(\frac{d}{dx}(f(x)-g(x)) = \frac{d}{dx}(f(x)) - \frac{d}{dx}(g(x))\). (Alternatively, for \(d(x)=f(x)-g(x)\), \(d'(x)=f'(x)-g'(x)\).)
- Constant Multiple Rule: \(\frac{d}{dx}(kf(x)) = k \frac{d}{dx}(f(x))\). (Alternatively, for \(m(x)=kf(x)\), \(m'(x)=kf'(x)\).)
Common Mistakes
- Misapplying the Power Rule: The Power Rule only applies to algebraic functions of the form \(y=x^n\). It does not apply to functions that raise a constant to a variable power, such as \(f(x)=3^x\). That is, \(\frac{d}{dx}(3^x) \neq x3^{x-1}\).