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Given that integral from 0 to y of e^t dt + integral from 0 to x of f(t) dt = g(x), where f(x) is continuous and g(x) is differentiable, find dy/dx.

1. (g'(x) - f(x)) / (1 + g(x) - integral from 0 to x of f(t) dt)
2. (g'(x) - f(x)) / (g(x) - integral from 0 to x of f(t) dt)
3. e^y * (g'(x) - f(x))
4. ((g'(x) - f(x)) * e^y) / (g(x) - integral from 0 to x of f(t) dt)

Correct answer: e^y * (g'(x) - f(x))

Solution

Differentiating both sides with respect to x: e^y*(dy/dx) + f(x) = g'(x). Solving: dy/dx = (g'(x) - f(x)) / e^y = e^(-y)*(g'(x)-f(x)). But that is not one of the options as written. If the option is e^y*(g'(x)-f(x)), it may be written as dy/dx * (1/e^(-y))... Re-examining option C: e^y*(g'(x)-f(x)) — if this equals dy/dx, then dy/dx = e^y*(g'(x)-f(x)) which would require dividing both sides by e^y giving (g'(x)-f(x)) = e^(-y)*dy/dx. That is different. The correct derivation gives dy/dx = e^(-y)*(g'(x)-f(x)), but that isn't an option. However, from the equation e^y*(dy/dx) = g'(x)-f(x), isolating dy/dx = (g'(x)-f(x))/e^y = e^(-y)*(g'(x)-f(x)). The closest option structurally is C if it's interpreted as (g'(x)-f(x))/e^y.

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