2.6.1: Resources and Key Concepts
- Page ID
- 192953
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Videos
- Derivative of exponential functions
- Derivative of \(e^x\)
- Derivative of \(b^x\)
- Derivative of hyperbolic functions
- Derivative of \( \sinh(x) \)
Key Concepts
Definitions
- The number \(e\): The irrational number defined as the limit \(\displaystyle \lim_{k \to \infty} \left(1 + \frac{1}{k}\right)^k = \lim_{n \to 0} (1+n)^{1/n}\).
Theorems
- Theorem: Derivative of \(e^x\): \(\frac{d}{dx}(e^x) = e^x\).
- Corollary: Derivative of \(b^x\): \(\frac{d}{dx}(b^x) = b^x \ln(b)\).
- Derivatives of Hyperbolic Functions:
- \(\frac{d}{dx}(\sinh x) = \cosh x\)
- \(\frac{d}{dx}(\cosh x) = \sinh x\)
- \(\frac{d}{dx}(\tanh x) = \text{sech}^2 x\)
- \(\frac{d}{dx}(\coth x) = -\text{csch}^2 x\)
- \(\frac{d}{dx}(\text{sech } x) = -\text{sech } x \tanh x\)
- \(\frac{d}{dx}(\text{csch } x) = -\text{csch } x \coth x\)
Common Mistakes
- Confusing \(\frac{d}{dx}(e^x)\) with \(\frac{d}{dx}(x^e)\): The rule for \(e^x\) (derivative is itself) is different from the Power Rule.
- Forgetting \(\ln(b)\) in \(\frac{d}{dx}(b^x)\): A common error is to write \(\frac{d}{dx}(b^x) = b^x\) instead of \(b^x \ln(b)\).
- Sign Errors in Hyperbolic Derivatives: Similar to trigonometric functions, some hyperbolic derivatives involve negative signs (\(\coth x, \text{sech } x, \text{csch } x\)). It's important to note that \(\frac{d}{dx}(\cosh x) = \sinh x\) (positive), unlike \(\frac{d}{dx}(\cos x) = -\sin x\).
- Incorrectly Applying Chain Rule: Forgetting to multiply by the derivative of the exponent when differentiating \(e^{u(x)}\) or \(b^{u(x)}\), or the argument of a hyperbolic function.