Final answer:
The expression simplifies to 2/(x^3) by using Leibniz rule and involves evaluating the function at t = x^2 and multiplying by the derivative of x^2.
Step-by-step explanation:
To simplify the expression involving the derivative of an integral, we can use the Leibniz rule, a specific form of differentiation under the integral sign. The expression provided is the derivative with respect to x of an integral from 0 to x2 of a function of t.
The Leibniz rule states that if we have:
d/dx ∫a(x)b(x) f(t, x) dt = f(b(x), x) · d(b(x))/dx - f(a(x), x) · d(a(x))/dx + ∫a(x)b(x) ∂f/∂x dt
In the given problem, a(x) = 0 and b(x) = x2, so the first part of the derivative of the integral simplifies as the function evaluated at t = x2 times the derivative of x2 due to the upper limit, minus the function evaluated at t = 0 times the derivative of 0 due to the lower limit.
Since the derivative of a constant is zero, the term involving the lower limit vanishes. Therefore, applying the Leibniz rule, we get:
d/dx ∫0x2 dt/t2 + 14 = 2x · (1/x4 + 14)
Finally, we simplify the right-hand side to get:
2/(x3)