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The Fundamental Theorem of Calculus is a cornerstone in the field of calculus that links differentiation with integration. It states that if a function is continuous over an interval, then integration and differentiation are inverse processes. In simple terms, if you have a function that is integrated over an interval and you take its derivative, you'll return to the original function.

This theorem is crucial in solving problems involving partial derivatives of integrals. For example, when we have a function like:\[f(x, y)=\int_{x}^{y}\left(t^{2}-1\right) dt\]we can find its derivative with respect to both variables thanks to this theorem.

Applying the theorem ensures that when you differentiate an integral, it leads directly to the integrand evaluated at the variable of differentiation. This principle works seamlessly, provided the function within the integral is continuous.

When we speak of partial derivatives, we're talking about finding the rate of change of a function concerning one of its variables, while all other variables are held constant. This is particularly useful in functions of several variables, where understanding how each variable independently affects the outcome is essential.

Consider the function:\[f(x, y)=\int_{x}^{y}\left(t^{2}-1\right) dt\]To find the partial derivative with respect to \(y\), you treat \(x\) as a constant and differentiate as if \(y\) were the only variable that changes. Applying the Fundatmental Theorem of Calculus, we derive \[\frac{\partial f}{\partial y} = y^2 - 1\]which directly evaluates the integrand at \(y\).

• This approach is direct.
• Makes use of the continuous nature of the function.

To find the derivative with respect to \(x\), \(y\) is held constant, and similar principles are applied to compute: \[\frac{\partial f}{\partial x} = -(x^2 - 1)\]Fundamental theorem ensures this process remains straightforward by tightly coupling differentiation and integration.

Integration limits play a vital role when differentiating integrals concerning their bounds. These limits define where the variable of integration starts and ends, and any change in them directly affects the derivative.

For the function: \[f(x, y)=\int_{x}^{y}\left(t^{2}-1\right) dt\]the limits are \(x\) and \(y\). When you take the partial derivative with respect to a limit, you essentially observe how changing that particular limit affects the integral's total value. Key points to consider:

• If the upper limit is the variable of differentiation, the derivative of the integral equals the integrand evaluated at that point.
• If the variable is at the lower limit, like \(x\), the derivative is the negative of the integrand evaluated at that point.

Thus, integrating from \(x\) to \(y\) and differentiating with respect to \(x\) yields a negative integrand result due to its position as the lower bound:\[\frac{\partial f}{\partial x} = -(x^2 - 1)\]This demonstrates how critical the role of integration limits are in finding and understanding partial derivatives, as they control the outcome of differentiation operations across limits.