15. Find f'(2) if f(x) = e^(g(x)) and g(x) = ∫(from 2 to x) t/(1+t⁴)dt.

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Identify the given functions: \( f(x) = e^{g(x)} \) and \( g(x) = \int_{2}^{x} \frac{t}{1+t^{4}} \, dt \). We need to find \( f'(2) \).

Recall that to find \( f'(x) \), we use the chain rule: \( f'(x) = e^{g(x)} \cdot g'(x) \). So, the next step is to find \( g'(x) \).

By the Fundamental Theorem of Calculus, the derivative of \( g(x) = \int_{2}^{x} \frac{t}{1+t^{4}} \, dt \) with respect to \( x \) is the integrand evaluated at \( x \): \( g'(x) = \frac{x}{1+x^{4}} \).

Substitute \( g'(x) \) back into the expression for \( f'(x) \): \( f'(x) = e^{g(x)} \cdot \frac{x}{1+x^{4}} \).

Finally, to find \( f'(2) \), evaluate \( g(2) = \int_{2}^{2} \frac{t}{1+t^{4}} \, dt \) (which is zero because the limits are the same), then substitute \( x=2 \) into \( f'(x) \) to get \( f'(2) = e^{g(2)} \cdot \frac{2}{1+2^{4}} \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Fundamental Theorem of Calculus

This theorem connects differentiation and integration, stating that if g(x) is defined as an integral with a variable upper limit, then g'(x) equals the integrand evaluated at x. Here, g(x) = ∫₂ˣ t/(1+t⁴) dt implies g'(x) = x/(1+x⁴).

Chain Rule

The chain rule is used to differentiate composite functions. For f(x) = e^(g(x)), the derivative f'(x) is e^(g(x)) multiplied by g'(x). This rule allows us to handle the exponential function with an inner function g(x).

Evaluating Derivatives at a Point

After finding the general derivative f'(x), evaluating it at a specific point (x=2) involves substituting x=2 into both f(x) and g'(x). This step provides the exact slope of the function at that point.