Please Help ASAP AP calculus Due in 1 day!!! Marking Brainliest!!!

1 answer:

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Answer:

(D) $\displaystyle \frac{12}{1 + ln8}$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Algebra I

Calculus

Derivatives

Derivative Notation

Basic Power Rule:

Derivative Rule [Chain Rule]: $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Integration

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

$\displaystyle f(x) = \int\limits^{x^3}_1 {\frac{1}{1 + ln(t)}} \, dt$

Step 2: Differentiate

Chain Rule: $\displaystyle f'(x) = \frac{d}{dx} \bigg[ \int\limits^{x^3}_1 {\frac{1}{1 + ln(t)}} \, dt \bigg] \cdot \frac{d}{dx}[x^3]$
Integration Rule - Fundamental Theorem of Calculus 2: $\displaystyle f'(x) = \frac{1}{1 + ln(x^3)} \cdot \frac{d}{dx}[x^3]$
Basic Power Rule: $\displaystyle f'(x) = \frac{1}{1 + ln(x^3)} \cdot 3x^{3 - 1}$
Simplify: $\displaystyle f'(x) = \frac{3x^2}{1 + ln(x^3)}$

Step 3: Evaluate

Substitute in x [Derivative]: $\displaystyle f'(2) = \frac{3(2)^2}{1 + ln(2^3)}$
Exponents: $\displaystyle f'(2) = \frac{3(4)}{1 + ln(8)}$
Multiply: $\displaystyle f'(2) = \frac{12}{1 + ln(8)}$