Final answer:
To find the derivative of the function f(x) defined by the integral, we apply Leibniz's rule, taking into account the dependency of the bounds and integrand on the variable x.
Step-by-step explanation:
The problem requires us to find the derivative of a function defined by a definite integral: f(x) = ∫₁ rac{dy}{x^2+y}. This can be approached using Leibniz's rule, a method for differentiating an integral with respect to a variable. According to this rule, when we have a definite integral that depends on a parameter, say x, and we want to differentiate it with respect to that parameter, we must take into account both the integral's bounds and its integrand.
To apply this rule to the given function, we need to differentiate under the integral sign and take the derivative of the upper bound with respect to x. The detailed process is as follows: first, differentiate the integrand with respect to x, while treating y as a constant; second, evaluate the integrand at the upper bound and multiply by the derivative of the upper bound; third, subtract the evaluation of the integrand at the lower bound (if the lower bound also depends on x).
In this case, the lower bound is a constant, so we only need to perform the first two steps. The resulting expression for the derivative of the function will depend on the value of x, as it is the upper limit of the integration.