91Ó°ÊÓ

Let's dive into the concept of taking the derivative of an integral, specifically how this works when dealing with a function of a new variable as an upper limit. Leibniz's rule for differentiation addresses this situation. It matches exactly the scenario when you have an integral with an upper boundary that is a function of the variable over which you are differentiating. When calculating the derivative of such an integral, the rule states that you take the integrand (the function inside the integral), substitute the upper limit for the variable of integration, and then multiply by the derivative of the upper limit with respect to the variable of interest.

In simpler terms, Leibniz's rule can be represented as:

• If you have: \[ \frac{d}{d x} \int_{a}^{f(x)} g(t) \, dt \]
• It simplifies to: \[ g(f(x)) \cdot \frac{d}{d x} f(x) \]

This concept allows us to simplify more complex expressions by considering only the effect of the changing boundary.

Understanding how boundaries affect an integral is essential in calculus, particularly when the boundary is itself a function. The integral's boundary can change dynamically, and in our exercise, the upper boundary is dependent on the variable \(x\).

Here, the boundary is \(\cos x\), which is not a constant, but rather a moving "target" depending on the value of \(x\). To apply Leibniz's rule, we have to compute the derivative of this boundary concerning \(x\). For \(\cos x\), this derivative is \(-\sin x\). Thus, every change in the integral boundary contributes to the entire derivative process; it reflects how much the shape of the region being integrated over changes as \(x\) varies.

Trigonometric functions play a crucial role in many calculus problems, especially when they form part of the function itself or its boundaries. In our example, \(\cos x\) emerges naturally as a boundary function in the integral setup.

When differentiating \(\cos x\) with respect to \(x\), we find \(-\sin x\). Recognizing how these trigonometric identities and derivatives interact is vital. They change the direction and rate of change in problems involving oscillating or rotating values.

In particular, remembering key differentiation rules such as:

• The derivative of \(\sin x\) is \(\cos x\)
• The derivative of \(\cos x\) is \(-\sin x\)

is imperative. This understanding helps in knowing how the trigonometric functions' integrative capacities influence the behavior of more complex mathematical expressions involved in computations like those found in calculus exercises.