Question

What is the derivative of x^x

Answer 1

`\displaystyle (d)/(dx)[x^x] = x^x[\ln (x) + 1]`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Product Rule]:
`\displaystyle (d)/(dx) [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)`

Derivative Rule [Chain Rule]:
`\displaystyle (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

Explanation:

Step 1: Define

Identify

`\displaystyle y = x^x`

Step 2: Differentiate

1. Rewrite:
   `\displaystyle y = e^\big{x\ln x}`
2. Exponential Differentiation [Derivative Rule - Chain Rule]:
   `\displaystyle y = e^\big{x\ln x} \cdot (d)/(dx)[x\ln x]`
3. Derivative Rule [Product Rule]:
   `\displaystyle y = e^\big{x\ln x}[(x)'\ln x + x(\ln x)']`
4. Basic Power Rule/Logarithmic Differentiation:
   `\displaystyle y = e^\big{x\ln x}(\ln x + 1)`

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

Unit: Differentiation