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

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

Examples

\textbf{Example One}

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

Source: http://quicklatex.com/cache3/5e/ql_8f6afcf624fd6ff3358d24064a1de55e_l3.png

 

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.

Source: http://quicklatex.com/cache3/3f/ql_1700b2158d1c78d7e22c228441b7633f_l3.png

 

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.

Source: https://www.flexiprep.com/NCERT-Exercise-Solutions/Mathematics/Class-11/posts/Ch-13-Limits-And-Derivatives-Exercise-13-2-Solutions-Part-6/Definition-of-derivative.png

 


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.

 

 

Source: http://quicklatex.com/cache3/53/ql_f80f9cba861c3cb5c43d293af3258953_l3.png

 


 

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.

Source: http://quicklatex.com/cache3/b9/ql_4a87bf6b238c8448d1068297300117b9_l3.png

 

 

\textbf{Example Two}

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

Source: http://quicklatex.com/cache3/d2/ql_7a21870fe1e85307f844c681695bb4d2_l3.png

 

\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.

Source: http://quicklatex.com/cache3/eb/ql_18a89af99eed896fe7d27a8f2d743ceb_l3.png

 

 

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