Question

Help.me with Ap calc please ​

Answer 1

`\displaystyle f'(1) = 6`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Explanation:

Step 1: Define

Identify

`\displaystyle f(x) = 5x - 1 + \ln x`

Step 2: Differentiate

1. [Function] Derivative Property [Addition/Subtraction]:
   `\displaystyle f'(x) = (d)/(dx)[5x] - (d)/(dx)[1] + (d)/(dx)[\ln x]`
2. Rewrite [Derivative Property - Multiplied Constant]:
   `\displaystyle f'(x) = 5 (d)/(dx)[x] - (d)/(dx)[1] + (d)/(dx)[\ln x]`
3. Derivative Rule [Basic Power Rule]:
   `\displaystyle f'(x) = 5 - 0 + (d)/(dx)[\ln x]`
4. Logarithmic Differentiation:
   `\displaystyle f'(x) = 5 + (1)/(x)`
5. Substitute in x:
   `\displaystyle f'(1) = 5 + (1)/(1)`
6. Simplify:
   `\displaystyle f'(1) = 6`

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

Unit: Differentiation