The Power Rule For Calculus Derivatives

A big part of calculus involves rates of change. Rates of changes involves the ratio of the differences in one variable when there is a change in another variable.

Calculus derivatives focus on the instantaneous rate of change at a point on the function’s domain. Given a step size or a small increment of h, as h approaches zero, the derivative at a point is obtained. This gives the limit definition of the derivative at a point a (in the function f(x)).

Limit Definition

    \[f'(a) = \displaystyle{\lim_{h \to 0}}\dfrac{f(a + h) - f(a)}{h}\]


\textbf{Example One}

With the limit definition of the derivative, find the derivative of g(x) = 3x at the point x = 2.



The derivative of g(x) = 3x at x = 2 is simply 3. This three represents the slope at the point x = 2. Furthermore, this slope of three represents the slope for all x-values in g(x).

\textbf{Example Two}

With the limit definition of the derivative, find the derivative of f(x) = -2x^2 for any value of x.



The derivative of f(x) = -2x^2 is f'(x) = -4x. In this second example, there is more algebra involved. The main goal is to use algebra and factoring to eliminate the h on the bottom. Once the h on the bottom of the fraction is removed, you can apply the limit as h approaches zero.



Power Rule

The limit definition of the derivative helps in understanding where the derivative comes from but it is not the fastest. A more efficient way and common way of obtaining derivatives is through the power rule for polynomials.

Consider the polynomial function f(x) where:

    \[f(x) = a_{n}x^{n} + a_{n - 1}x^{n - 1} + ... + a_{3}x^3 + a_{2}x^2 + a_{1}x + a_{0}\]

where a_{n} to a_{0} are numeric constant coefficients.

The derivative f'(x) of the above polynomial function would be:

    \[f'(x) = a_{n}n x^{n - 1} + a_{n - 1} (n - 1) x^{n - 2} + ... + 3 a_{3}x^2 + 2 a_{2}x^1 + a_{1}\]

Notice that the a_{0} intercept term goes away. In addition, the terms are multiplied by the exponent and the exponent decreases by one. The examples below will give a better idea on how this all works.

Addition, Subtraction Rules For Derivatives

For calculus derivatives, they can operate with addition and subtraction signs. Here are some basic rules.






Derivatives Examples

Here are some examples of finding derivatives of functions. Note that f'(x) refers to the same derivative as \dfrac{d}{dx} f(x).

\textbf{Example One}

Find the derivative of the function f(x) = 2x.




\textbf{Example Two}

Given f(x) = 3x^3 what is f'(x)?



\textbf{Example Three}

What is the derivative of f(x) = x^{\pi}?

    \[f'(x) =  \pi x^{\pi - 1}\]

Remember that \pi \approx 3.14 is a number.

\textbf{Example Four}

What is the derivative of g(x) = x(x^2 - 4x + 2)?

This particular example looks scary but it is actually not too bad. The distributive law is applied first and derivatives can be taken separately.

Rewrite g(x) as:

    \[g(x) = x(x^2 - 4x + 2) = x^3 - 4x^2 + 2x\]

Then you can take the derivatives of each term.




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