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An antiderivative is a function that reverses the process of differentiation. When you find the antiderivative of a function, you are essentially working out the indefinite integral. In simple terms, the antiderivative represents the original function before it was differentiated. For instance, if you have a function \(f(x) = 5 - x\), finding its antiderivative involves integrating it to get back the original function from which \(f(x)\) was derived.The antiderivative is always associated with a constant of integration, denoted as \(C\). This constant represents the whole family of functions that could have the given derivative, since any constant becomes zero when differentiated. Thus, the indefinite integral of \(5-x\) is \(5x - \frac{1}{2}x^{2} + C\). Here, the terms \(5x\) and \(-\frac{1}{2}x^{2}\) are the components adjusted after integrating each part of the function separately.

Differentiation is the process of finding the derivative of a function. The derivative represents the rate of change or the slope of the function at any given point. Checking an integral solution by differentiation is a vital part of the calculation, as it verifies whether the indefinite integral was computed correctly.For our function \(5x - \frac{1}{2}x^{2} + C\), applying differentiation allows us to backtrack to the original function. When we differentiate:

• The derivative of \(5x\) is 5.
• The derivative of \(-\frac{1}{2}x^{2}\) is \(-x\).
• The constant \(C\), when differentiated, becomes 0.

This differentiation brings us back to the original function \(5 - x\), confirming that our antiderivative was correctly found.

The power rule is a key concept for both differentiation and integration. It's a formula that makes it easier to find the derivatives or integrals of power functions. For differentiation, the power rule states that if you have a function \(x^n\), the derivative is \(nx^{n-1}\).Similarly, for integration, which is essentially the reverse of differentiation, the power rule provides a simple formula to find the antiderivative. It states that the integral of \(x^n\) is \(\frac{1}{n+1}x^{n+1} + C\). In practice, this means:

• For differentiating \(x^2\), you get \(2x\) because you multiply the exponent by the coefficient and subtract one from the exponent.
• For integrating \(x\), you apply the reverse power rule to find \(\frac{1}{2}x^{2}\), adding one to the exponent and dividing by the new exponent.

Mastering the power rule simplifies the process of both finding antiderivatives and confirming them through differentiation.