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

Verified Answer :

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Answer:

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

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Integration

  • Integrals

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

Step-by-step 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

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