7. If $f(x)=\int_{1}^{x^{3}} \frac{1}{1+\ln t} d t$ for $x \geq 1$, then $f^{\prime}(2)=$
(A) $\frac{1}{1+\ln 2}$
(B) $\frac{12}{1+\ln 2}$
(C) $\frac{1}{1+\ln 8}$
(D) $\frac{12}{1+\ln 8}$

Q: 7. If $f(x)=\int_{1}^{x^{3}} \frac{1}{1+\ln t} d t$ for $x \geq 1$, then $f^{\prime}(2)=$ (A) $\frac{1}{1+\ln 2}$ (B) $\frac{12}{1+\ln 2}$ (C) $\frac{1}{1+\ln 8}$ (D) $\frac{12}{1+\ln 8}$

∻Solution by Steps∻ ‖ step 1 ⋮ To find $ f'(x) $, we use the Fundamental Theorem of Calculus Part 1, which states that if $ F(x) = \int_{a}^{g(x)} f(t) dt $, then $ F'(x) = f(g(x)) \cdot g'(x) $... ‖ ‖ step 2 ⋮ Applying this to $ f(x) = \int_{1}^{x^3} \frac{1}{1+\ln t} dt $, we get $ f'(x) = \frac{1}{1+\ln(x^3)} \cdot \frac{d}{dx}(x^3) $... ‖ ‖ step 3 ⋮ The derivative of $ x^3 $ is $ 3x^2 $, so $ f'(x) = \frac{3x^2}{1+\ln(x^3)} $... ‖ ‖ step 4 ⋮ To find $ f'(2) $, we substitute $ x = 2 $ into the derivative to get $ f'(2) = \frac{3(2)^2}{1+\ln(2^3)} $... ‖ ‖ step 5 ⋮ Simplifying, we find $ f'(2) = \frac{12}{1+\ln 8} $... ‖ ***D*** ∻Key Concept∻ ⚹Fundamental Theorem of Calculus Part 1⚹ ∻Explanation∻ ⚹The derivative of the integral from a constant to a function of $ x $ is the integrand evaluated at the function times the derivative of the function.⚹◊What is the derivative of the function $f(x)$ at $x=2$?⍭ Generate me a similar question◊

Question

7. If $f(x)=\int_{1}^{x^{3}} \frac{1}{1+\ln t} d t$ for $x \geq 1$, then $f^{\prime}(2)=$
(A) $\frac{1}{1+\ln 2}$
(B) $\frac{12}{1+\ln 2}$
(C) $\frac{1}{1+\ln 8}$
(D) $\frac{12}{1+\ln 8}$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find $ f'(x) $, we use the Fundamental Theorem of Calculus Part 1, which states that if $ F(x) = \int_{a}^{g(x)} f(t) dt $, then $ F'(x) = f(g(x)) \cdot g'(x) $... ‖ ‖ step 2 ⋮ Applying this to $ f(x) = \int_{1}^{x^3} \frac{1}{1+\ln t} dt $, we get $ f'(x) = \frac{1}{1+\ln(x^3)} \cdot \frac{d}{dx}(x^3) $... ‖ ‖ step 3 ⋮ The derivative of $ x^3 $ is $ 3x^2 $, so $ f'(x) = \frac{3x^2}{1+\ln(x^3)} $... ‖ ‖ step 4 ⋮ To find $ f'(2) $, we substitute $ x = 2 $ into the derivative to get $ f'(2) = \frac{3(2)^2}{1+\ln(2^3)} $... ‖ ‖ step 5 ⋮ Simplifying, we find $ f'(2) = \frac{12}{1+\ln 8} $... ‖ ***D*** ∻Key Concept∻ ⚹Fundamental Theorem of Calculus Part 1⚹ ∻Explanation∻ ⚹The derivative of the integral from a constant to a function of $ x $ is the integrand evaluated at the function times the derivative of the function.⚹◊What is the derivative of the function $f(x)$ at $x=2$?⍭ Generate me a similar question◊

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