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The power rule is a basic differentiation rule used in calculus. It allows us to find the derivative of a function that is a power of x. For a function of the form \(f(x) = x^n\), where n is any real number, the power rule states: \[ \frac{d}{dx} [x^n] = nx^{n-1} \] This means that we multiply the original power by the coefficient and then reduce the power by one.

For example, if we have a function \(y = x^5\), applying the power rule would yield the derivative \(y' = 5x^4\).

In our given exercise, the function is \(y = 2x^{15}\). Applying the power rule, we take the exponent 15, multiply it by the coefficient 2, and then reduce the exponent by one. This gives us: \[ \frac{d}{dx} [2x^{15}] = 15 \times 2x^{15-1} = 30x^{14} \] Thus, the derivative of \(y = 2x^{15}\) is \(30x^{14}\).

A derivative represents the rate at which a function is changing at any given point. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes slightly. The notation for the derivative of a function \(y\) with respect to \(x\) is \(\frac{dy}{dx}\).

Geometrically, the derivative of a function at a point is the slope of the tangent line to the graph of the function at that point.

For the given function \(y = 2x^{15}\), the derivative tells us how the value of \(y\) changes as \(x\) changes. Using the power rule, we found that \(\frac{dy}{dx} = 30x^{14}\). This means that for small changes in \(x\), the changes in \(y\) can be approximated by multiplying \(x^{14}\) by 30.

Understanding derivatives is crucial for solving problems related to rates of change in various fields such as physics, economics, and biology.

In mathematics, a function is a relationship between a set of inputs and a set of possible outputs such that each input is related to exactly one output.

Functions can be represented in various forms including equations, graphs, tables, or words.

One of the key aspects of functions is that they describe how one quantity changes in relation to another. For example, the function \(y = 2x^{15}\) describes how \(y\) changes as \(x\) changes.

When dealing with functions, it's important to understand their different properties like domains, ranges, and whether they are continuous or discrete.

Calculus often focuses on the analysis of functions, particularly through differentiation and integration, to understand their behavior comprehensively.