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Understanding the derivative of a function is essential in calculus and helps to describe the rate at which one quantity changes with respect to another. In simple terms, the derivative of a function at a point measures the rate at which the function's output changes as its input changes. It's represented as either \( f'(x) \) or \( \frac{d}{dx}f(x) \).

The process of finding the derivative is known as differentiation. There are various rules to differentiate functions, and the power rule is among the most fundamental. When we say a function is 'increasing everywhere', it means that for any two points \( a \) and \( b \) on the function where \( a < b \) the function's output at \( b \) is greater than its output at \( a \)---this corresponds to a positive derivative at all points on the function.

Understanding Critical Points:
Critical points of a function can provide us with valuable information about its behavior. A critical point occurs where the derivative \( f'(x) \) is zero or undefined. These points are possible locations for local maxima, minima, or inflection points. To find critical points, one must set the derivative equal to zero and solve for \( x \) or determine where the derivative does not exist. In the context of increasing functions, if a function does not have any critical points, or if all critical points lead to positive slopes, this suggests that the function could be increasing everywhere.

Exploring the Discriminant:
In algebra, the discriminant of a quadratic equation of the form \( ax^2 + bx + c = 0 \) is given by \( D = b^2 - 4ac \). It determines the nature of the roots of the equation—the solutions for \( x \).

• If \( D > 0 \) there are two distinct real roots.
• If \( D = 0 \) there is exactly one real root (a repeated root).
• If \( D < 0 \) there are no real roots, only complex roots.

When applying this concept to the derivative of a function—as in the given problem—the discriminant can reveal whether critical points exist in the real number system. If the discriminant is negative, like in the example, there are no real roots to the derivative equation, suggesting that there are no critical points and, therefore, the function may be monotonic (always increasing or always decreasing).

The Power of the Power Rule:
One of the most utilized shortcuts in taking derivatives is the power rule. The power rule states that for any real number \( n \) the derivative of \( x^n \) with respect to \( x \) is \( nx^{n-1} \). Applying this rule makes finding the derivative of any term involving a power of \( x \) straightforward and efficient.

For instance, in our given function \( f(x) = 2x^5 + x^3 + 2x \) using the power rule, we differentiate each term separately: the derivative of \( 2x^5 \) is \( 10x^4 \) using the power rule—and similar steps are taken for each term involving a power of \( x \) to get the complete derivative. Mastery of the power rule is indispensable in calculus, particularly when dealing with polynomials and proving if functions are increasing or decreasing.