91Ó°ÊÓ

The Chain Rule is a fundamental tool in calculus used to differentiate compositions of functions. In simple terms, it helps us find the derivative of a function nested inside another function. For instance, when we have a function of the form \( f(g(x)) \), we differentiate it by first finding the derivative of the outer function \( f \), and then multiplying it by the derivative of the inner function \( g \). This is expressed mathematically as:\[\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)\]In the context of the given exercise, the use of the Chain Rule is critical. We have the integral of \( \sin(t^2) \) where the upper limit is a function of \( x \), specifically \( \sqrt{x} \). When differentiating, the Chain Rule tells us to find the derivative of the sine function evaluated at \( \sqrt{x}^2 \), and then multiply this by the derivative of the upper limit \( \sqrt{x} \). This step is crucial in arriving at the correct solution for \( \frac{dy}{dx} \).

The differentiation of integrals, specifically those with variable limits, is an important concept in calculus. The Fundamental Theorem of Calculus is key here. It connects integration and differentiation, providing a method for finding derivatives of integrals. When dealing with an integral like \( y = \int_{a}^{g(x)} f(t) \, dt \), the theorem states that the derivative of \( y \) with respect to \( x \) is:\[\frac{dy}{dx} = f(g(x)) \cdot g'(x)\]This indicates that we evaluate the function \( f(t) \) at \( g(x) \) and then multiply by the derivative of \( g(x) \). In the provided exercise, this theorem helps transition from the integral representation of \( y \) to finding \( \frac{dy}{dx} \), which involves the sine function and the upper limit \( \sqrt{x} \). The swapped limits initially suggest a negative sign, confirmed by the theorem's application, making sure directionality is respected in the integration bounds.

Variable limits of integration occur when the limits of an integral are functions of another variable, often \( x \). This situation frequently arises in problems involving definite integrals in which integration bounds depend on variables. To handle these effectively, one must correctly apply the Fundamental Theorem of Calculus and strategies like the Chain Rule.In the given exercise, the limit \( \sqrt{x} \) changes how we approach the problem. We need to take into account that reversing the limits of integration multiplies the integral by \(-1\). This is evident when we rewrite \( y = \int_{\sqrt{x}}^{0} \, \sin(t^2) \, dt \) as \( y = -\int_{0}^{\sqrt{x}} \sin(t^2) \, dt \).After reversing the limits and acknowledging the variable dependence, differentiation using the Fundamental Theorem leads to an expression that involves both the function evaluated at \( g(x) \) and the derivative \( g'(x) \), ultimately simplifying the calculation of \( \frac{dy}{dx} \). These fundamentals are essential for solving integrals with variable limits efficiently.