2.8.1: Resources and Key Concepts
- Page ID
- 192961
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Videos
- Derivative of logarithmic functions
- Derivative of function to a functional power
Key Concepts
Definitions
- Logarithmic Differentiation: A technique used to differentiate functions, particularly those of the form \(h(x) = g(x)^{f(x)}\) or complex products/quotients, by first taking the natural logarithm of both sides of \(y=h(x)\), using properties of logarithms to simplify, then differentiating implicitly with respect to \(x\), and finally solving for \(\frac{dy}{dx}\).
Theorems
- Theorem: The Derivative of the Natural Logarithmic Function: If \(y = \ln x\), then \(\frac{dy}{dx} = \frac{1}{x}\).
- Theorem: The General Derivative of a Logarithmic Function: \(\frac{d}{dx}(\log_b (x)) = \frac{1}{x \ln(b)}\).
Common Mistakes
- Incorrectly Applying Logarithm Properties: Errors in using logarithm properties (e.g., \(\ln(A+B) \neq \ln A + \ln B\)) before differentiation can lead to incorrect derivatives.
- Errors in Applying Logarithm Properties: Incorrectly expanding or simplifying the logarithmic expression (e.g., \(\ln(A+B) \neq \ln A + \ln B\)) before differentiation.
- Forgetting to Differentiate Implicitly: After taking \(\ln y\), forgetting that its derivative with respect to \(x\) is \(\frac{1}{y}\frac{dy}{dx}\), not just \(\frac{1}{y}\).
- Not Multiplying by \(y\) at the End: After finding \(\frac{1}{y}\frac{dy}{dx} = \text{expression}\), forgetting to multiply the "expression" by \(y\) (and substituting back the original function for \(y\)) to solve for \(\frac{dy}{dx}\).
- Applying Logarithmic Differentiation When it Cannot Be Applied: While it might seem to be attractive to use logarithmic differentiation for any function that contains compositions of powers of rational functions, you must remain aware that the argument of a logarithm has to be positive. Therefore, using logarithmic differentiation to find the derivative of, for example, \( y = \sqrt[3]{\frac{(x-2)^3}{x(x+8)^2}} \) would not work because the radicand could be negative (we can take odd roots of negative numbers).