One of the fundamental questions in calculus is how fast something is changing at a particular moment in time. For example, consider the price of a stock on the New York Stock Exchange. Before calculus, algebraically we could determine the value of a stock at a certain moment in time in the past. Now, using calculus techniques, we can determine how fast this stock price is changing at a specific moment in time. In calculus terminology we refer to this as the derivative function. Graphically, the derivative evaluated at a point is the slope of the tangent line to the function at that point. 

Given a polynomial which is a power function of the form:

$f(x)=axn$ where $a$, $n \in \mathbb R$

The derivative is denoted $f′(x)$ and can be written as: $f′(x)=nax^{n - 1}$ 

This is referred to as The Power Rule in calculus and is used in determining the instantaneous rate of change of any power function at a specific moment in time.

Example 1: Determine $f′(x)$ of the function $f(x)=2x^3+4x^2−10x+1$

Using the Power Rule from above, we can differentiate term by term to obtain:

$f′(x)=(2)(3)x^{3-1}+(4)(2)x^{2-1}$ $-$ $10(1)x^{1-0} + 1(0)x^{0-1}$
$f′(x)= 6x^2+8x-10$

This function, the derivative function $f′(x)$, will tell us how fast or slow the original function $f(x)$ is changing at any specific moment in time $x$.

Note: The derivative of the constant $+1$ in the function is $0$. This is not surprising since constants do not change.

Example 2: Determine the slope of the tangent line to the function $y=−x^3+2x−1$ at the point where $x=2$.

The change in $y$ with respect to $x$, or the derivative of $y$, is written as $\dfrac{dy}{dx}$. Using the Power Rule from above, we can differentiate term by term:

$\dfrac{dy}{dx} = (−1)(3)x^{3−1} + (2)(1)x^{1−1}$
$\dfrac{dy}{dx}= −3x^2 + 2$

$\dfrac{dy}{dx}|_{x=2} = −3(2)^2+2$
$\dfrac{dy}{dx}|_{x=2}=−12+2$
$\dfrac{dy}{dx}|_{x=2}=−10$

Note: This value is telling us that at the moment in time when $x=2$, the function is changing at a rate of $−10$ units per unit of time.