If f(x) = e^{g(x)} and g(x) = int_{2}^{x} fra | vedclass

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Question

(A) Given $f(x) = e^{g(x)}$.By differentiating with respect to $x$,we get $f'(x) = e^{g(x)} \cdot g'(x)$.Using the Leibniz integral rule,$g'(x) = \fra

Vedclass · Accepted Answer

$2/17$

Vedclass · Answer

(A) Given $f(x) = e^{g(x)}$.By differentiating with respect to $x$,we get $f'(x) = e^{g(x)} \cdot g'(x)$.Using the Leibniz integral rule,$g'(x) = \frac{d}{dx} \int_{2}^{x} \frac{t}{1 + t^4} \, dt = \frac{x}{1 + x^4}$.Now,evaluate $g(2) = \int_{2}^{2} \frac{t}{1 + t^4} \, dt = 0$.Therefore,$f'(2) = e^{g(2)} \cdot g'(2) = e^0 \cdot \frac{2}{1 + 2^4} = 1 \cdot \frac{2}{1 + 16} = \frac{2}{17}$.

If $f(x) = e^{g(x)}$ and $g(x) = \int_{2}^{x} \frac{t}{1 + t^4} \, dt$,then $f'(2)$ has the value equal to:

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