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The Fundamental Theorem of Calculus is a cornerstone concept in calculus that links the concept of differentiation with that of integration. It has two main parts, but here we focus on how it helps to differentiate an integral over a function of a variable.

• Part 1: It states that if you have a continuous function on an interval, the function's integral over that interval can be differentiated to obtain the original function.
• Part 2: It tells us that the derivative of an integral from a to x of f(t) dt is simply f(x).

In practice, this theorem provides a powerful tool for differentiating integrals with variable limits, like in our example exercise. When differentiating a definite integral, we often apply what's known as the Leibniz Rule, an extension of the theorem for integrals with variable limits.

Here, when the upper limit of integration is a function of t, like \( \sqrt{t} \), we use it directly inside the integrand, multiplying by the derivative of the upper limit. This simplifies the process of differentiation considerably, avoiding lengthy calculations. Understanding this concept is crucial for moving seamlessly between integration and differentiation, as demonstrated by the steps in solution.

A definite integral is a key concept representing the accumulation of quantities, such as area under a curve, over an interval \([a, b]\). Unlike an indefinite integral, it does not include a constant of integration because the result is a real number.

In solving the given exercise, we first evaluated a definite integral from 0 to \( \sqrt{t} \). This process provided the groundwork for finding a function that needed to be differentiated eventually. Here’s how it typically works:

• Setup: Identify the limits and the function inside the integral.
• Evaluate: Substitute the upper limit into the indefinite integral result, then subtract the result of substituting the lower limit.

This evaluation leads us to an expression involving \( t \) that we can later differentiate to find the derivative. This step is crucial because if we directly differentiate the integrand itself without considering the limits, we can make errors in results. By properly carrying out the definite integral, we prepare ourselves for an accurate application of the Fundamental Theorem of Calculus.

The Chain Rule is a fundamental technique in calculus used to differentiate composite functions. When a function is nested within another function, the Chain Rule provides a systematic method to find the derivative.

For example, in the differentiation of \( 3 \arcsin(\sqrt{t}) \), we utilize the Chain Rule because both the arcsine function and the square root function are involved. Generally, the Chain Rule is applied as follows:

• Outer Derivative: Take the derivative of the outer function first.
• Inner Derivative: Multiply by the derivative of the inner function.

In our scenario, we first differentiate \( 3 \arcsin(u) \) where \( u = \sqrt{t} \), resulting in \( 3 / \sqrt{1-u^2} \). Then we multiply by the derivative of \( \sqrt{t} \), which is \( 1/(2\sqrt{t}) \).

Thus, correctly using the Chain Rule ensures we accommodate eventual changes both inside and outside the main function, giving us the final derivative needed in our integral differentiation solution.