Question

Find the second derivative of log x

Answer 1

`\displaystyle y'' = (-1)/(\ln(10)x^2)`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Explanation:

Step 1: Define

Identify

`\displaystyle y = \log (x)`

Step 2: Differentiate

1. Logarithmic Differentiation:
   `\displaystyle y' = (1)/(\ln(10)x)`
2. Derivative Property [Multiplied Constant]:
   `\displaystyle y'' = (1)/(\ln 10) (d)/(dx) \bigg[ (1)/(x) \bigg]`
3. Basic Power Rule:
   `\displaystyle y'' = (-1)/(\ln(10)x^2)`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation