91Ó°ÊÓ

The chain rule is a fundamental concept for finding derivatives in calculus. It is used when you have a composition of functions. Suppose you have two functions, say \( u = g(x) \) and \( y = f(u) \). The chain rule tells us how to differentiate \( y \) with respect to \( x \). Essentially, it states that:

• \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \)

To put it simply, you take the derivative of the outer function with respect to the inner function and multiply that by the derivative of the inner function with respect to \( x \).
In the original exercise, the function \( \sin^3 x \) was differentiated using the chain rule. First, take the derivative of \( u^3 \) with \( u = \sin x \), which is \( 3u^2 \). Then, multiply it by the derivative of \( \sin x \), which is \( \cos x \). The result is \( 3\sin^2 x \cdot \cos x \). This application of the chain rule shows how it breaks down complex derivatives into manageable parts.

Leibniz's rule is used for differentiating an integral with variable limits. It's a powerful tool, especially when the limits of integration depend on \( x \), because it accounts for the changes in both the boundaries and the integrand. The rule is expressed as:

• \( \frac{d}{dx} \int_{a(x)}^{b(x)} f(t) \, dt = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x) \)

This formula tells us to evaluate the integrand at the upper bound, multiply by the derivative of that upper bound, and subtract the integrand evaluated at the lower bound, multiplied by the derivative of that lower bound.
In the given problem, the upper limit is \( \sin x \) and the lower limit \( a(x) = 1 \) is constant (hence its derivative is 0). Substituting into Leibniz's rule, you find \( f(t) = 3t^2 \) evaluated at \( b(x) = \sin x \), which gives \( 3(\sin x)^2 \). Multiply this by the derivative of the upper bound, \( \cos x \), resulting in \( 3\sin^2 x \cdot \cos x \). This connects the application of calculus to real-world scenarios where limits might not be constant.

A definite integral is the sum of values of a function \( f(t) \) on an interval \( [a, b] \). It is represented as \( \int_a^b f(t) \, dt \). The boundaries \( a \) and \( b \) are called the limits of integration, where \( a \) is the lower limit and \( b \) is the upper limit.
When you evaluate a definite integral, you first find the antiderivative of the function, then apply the Fundamental Theorem of Calculus, which states:

• \[ \int_a^b f(t) \, dt = F(b) - F(a) \]

where \( F(t) \) is the antiderivative of \( f(t) \).
In our exercise, the definite integral \( \int_{1}^{\sin x} 3t^2 \, dt \) is calculated by finding the antiderivative \( t^3 \) and evaluating it between the limits \( t = 1 \) and \( t = \sin x \). The result is \( (\sin x)^3 - 1 \). This understanding helps bridge the process of integration to other differential operations.

Differentiating under the integral sign is a strategy used to find the derivative of an integral when the limits depend on \( x \). This method is particularly useful when direct integration is difficult, or the integrand contains variable limits. It's based on Leibniz's rule.
To understand it better, consider the exercise: it required differentiating the integral \( \int_{1}^{\sin x} 3t^2 \, dt \) directly. Here, the derivative under the integral is managed smoothly by applying Leibniz's rule, leading to the derivative \( 3(\sin x)^2 \cdot \cos x \).

• This technique allows seeing the interaction between integration and differentiation and illustrates how calculus can elegantly solve complex problems beyond basic integrals or derivatives.

The method highlights the flexibility of calculus in handling variable limits, showcasing a depth of calculus applications often required in physics and engineering to model dynamic systems.