91Ó°ÊÓ

The idea of an integral is foundational in calculus. Integrals allow us to determine the total accumulation of quantities, whether that be areas under curves, total distance traveled, or other types of sums. The process of integrating is essentially the reverse of differentiating.
In this problem, the integral function \( F(x) = \int_{0}^{x^2} \frac{3}{t+1} dt \) represents the area under the curve from \(t = 0\) to \(t = x^2\) for the curve defined by \(\frac{3}{t+1}\).
Integrals can be definite, like the one in this exercise, where we calculate an actual numeric value or an expression for the total area. Understanding how to take the derivative of such an integral relies on the Fundamental Theorem of Calculus.

Derivatives signify the rate of change and tell us how a function changes at any given point.
In simpler terms, while integrals accumulate quantities, derivatives break them down to small, instantaneous changes.

• The function \( F'(x) \) describes how \( F(x) \) itself changes with \( x \).
• In this exercise, we find the derivative \( F'(x) \) by using the chain rule, which is critical when dealing with composition of functions like the upper limit \( x^2 \).

Differentials, as calculated here using \( \frac{d}{dx}(x^2) = 2x \), show us precisely how changes in \( x \) impact \( F(x) \). Thus, the derivative \( F'(x) = \frac{6x}{x^2+1} \) tells us how the integral value, \( F(x) \), changes when \( x \) changes.

The chain rule is a powerful method in calculus for differentiating composite functions. Whenever you have a function nested within another, the chain rule is your go-to tool.
In symbols, if you have a function \( g(x) \) inside another function \( f(u) \), where \( u = g(x) \), the derivative \( \frac{d}{dx}f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \).

• In our problem, we see a classic chain rule scenario: the integral’s upper limit is \( x^2 \) which is a function of \( x \).
• So, we first replace the variable of integration in the integrand with the upper limit to form \( \frac{3}{x^2+1} \).

Then the chain rule states we would multiply this result by the derivative of the upper limit, i.e., \( \frac{d}{dx}(x^2) = 2x \). This is how we arrived at the final expression for the derivative: \( \frac{6x}{x^2+1} \).
Understanding the application of the chain rule in such contexts allows us to differentiate more complex functions successfully.