Question

Find f'(0.3) for f(x) = the definite integral, bottom number 0, top number x, of arccos(t)dt

Answer 1

`\displaystyle f'(0.3) = \arccos (0.3)`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Integration

• Integrals

Integration Rule [Fundamental Theorem of Calculus 2]:
`\displaystyle (d)/(dx)[\int\limits^x_a {f(t)} \, dt] = f(x)`

Explanation:

Step 1: Define

Identify

`\displaystyle f(x) = \int\limits^x_0 {\arccos (t)} \, dt`

Step 2: Differentiate

1. Integration Rule [Fundamental Theorem of Calculus 2]:
   `\displaystyle f'(x) = \arccos (x)`

Step 3: Evaluate

1. Substitute in x [Function]:
   `\displaystyle f'(0.3) = \arccos (0.3)`

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

Unit: Integration