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3.6

Derivatives of Logarithmic Functions

Derivatives of Logarithmic Functions (Various Formulas)

$$\begin{gathered} \frac{d}{dx}\left({\log }_{a}x\right) = \frac{1}{x \ln a} \\ \frac{d}{dx}(\ln x) = \frac{1}{x} \\ \frac{d}{dx}(\ln u)= \frac{1}{u}\frac{du}{dx} \\ \text{or} \\ \frac{d}{dx}\left[\ln \left(g\left(x\right)\right)\right] =\frac{g\text{'}\left(x\right)}{g\left(x\right)} \\ \frac{d}{dx}\ln \left|x\right| = \frac{1}{x} \end{gathered}$$

Steps in Logarithmic Differentiation:

1. Take natural logarithms of both sides of an equation y = f(x) and use the Laws of Logarithms to simplify.
2. Differentiate implicitly with respect to x.
3. Solve the resulting equation for y'.

The Number e as a Limit

$$\begin{gathered} e = \underset{x\to 0}{\lim }(1+x{)}^{\frac{1}{x}} \\ e = \underset{n\to \infty }{\lim }(1+\frac{1}{n}{)}^n \end{gathered}$$

If f(x) = ln x, then f'(x) = 1/x. Thus f'(1) = 1. We now use this fact to express the number e as a limit above. If we put n = 1/x in the first formula, then n→∞ as x→0+, and so an alternative expression is the second above with n approaching infinity.

Links

• Home
• 3.1: Derivatives of Polynomials and Exponential Functions
• 3.2: The Product and Quotient Rules
• 3.3: Derivatives of Trigonometric Functions
• 3.4: The Chain Rule
• 3.5: Implicit Differentiation
• 3.6: Steps in Logarithmic Differentiation
• 3.7: Rates of Change in the Natural and Social Sciences
• 3.8: Exponential Growth and Decay
• 3.9: Related Rates
• 3.10: Linear Approximations and Differentials
• 3.11: Hyperbolic Functions