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The power rule is an essential principle in calculus, especially when it comes to differentiating polynomials. It simplifies the process of finding the rate of change of a function and is expressed as follows: if you have a function of the form f(x) = x^n where n is any real number, the derivative of this function with respect to x is f'(x) = nx^(n-1).

When you apply the power rule to the given function f(x) = x^2 + x, you treat each term separately. For the x^2 term, using the power rule, the exponent 2 is brought down to multiply the base x, followed by decreasing the exponent by 1. This results in 2x^(2-1), which simplifies to 2x. For the second term, x, it is essentially x^1, so applying the power rule gives us its derivative as 1x^(1-1), which simplifies to 1 since any number raised to the 0 power is 1.

Thus, using the power rule, the derivative of the original function is f'(x) = 2x + 1. This rule is widely applicable and makes differentiation a much more manageable task, but it's crucial to remember that the power rule only applies when the exponent is a real number and the base is x.

When discussing the existence of derivatives, we are essentially asking if it's possible to determine the instantaneous rate of change of a function at any point within its domain. For the function in question, being continuous and smooth over the entire real number line is indicative of this possibility.

Linear functions and polynomials are prime examples of functions whose derivatives exist everywhere within their domain, which includes all real numbers. They do not have any 'kinks' or 'sharp turns,' and they do not exhibit any discontinuities or vertical tangents. Since the derivative of a function at a point gives us the slope of the tangent line at that point, for the function f(x) = x^2 + x, we can confidently say that its derivative f'(x) = 2x + 1 exists for every real number x.

The justification lies in recognizing that the function being differentiated is a smooth curve without any jumps or undefined regions. To further cement this concept, it can be beneficial to graphically represent the function and observe how the tangent line smoothly touches the curve at any selected x value.

Linear functions are foundational in algebra and calculus due to their straightforward properties and representation. A linear function is of the form f(x) = mx + b, where m is the slope and b is the y-intercept. These functions graph as straight lines, which is why they are termed 'linear'.

One significant aspect of linear functions is their differentiability. Since they do not have breaks, bends, or cusps, the slope of a linear function is consistent along the entire line. This consistency means that if you were to find the derivative of a linear function, you would get a constant value. For the derivative f'(x) = 2x + 1, obtained from differentiating f(x) = x^2 + x, it simplifies even further into the format of a linear function, f'(x) = 2x + 1, where the slope is 2 and the y-intercept is 1.

Understanding linear functions is crucial for interpreting derivatives since the derivative itself often represents how a function is changing at any given point. For f'(x), this change is steady and does not fluctuate, confirming the continuity and existent derivative of the original function over the entire domain of real numbers.