Question

What's the derivative of tan^-1 (lnx) ...?

Answer 1

`\displaystyle (dy)/(dx) = (1)/(x \Big( (\ln x)^2 + 1 \Big))`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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

Explanation:

Step 1: Define

Identify

`\displaystyle y = \arctan (\ln x)`

Step 2: Differentiate

1. Trigonometric Differentiation [Derivative Rule - Chain Rule]:
   `\displaystyle y' = ((\ln x)')/((\ln x)^2 + 1)`
2. Logarithmic Differentiation:
   `\displaystyle y' = ((1)/(x))/((\ln x)^2 + 1)`
3. Simplify:
   `\displaystyle y' = (1)/(x \Big( (\ln x)^2 + 1 \Big))`

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

Unit: Differentiation