In this sub-topic, we will explore the concept of derivatives and how they can be used to understand and analyse functions of the form $ax^n$, where $a$ is any rational constant and $n$ is a positive integer or zero.

What is a Derivative?

The derivative of a function is a measure of how quickly that function changes at a specific point. It tells us the slope of the tangent line to the function's graph at that point.

Derivatives of $ax^n$

The derivative of a function of the form $ax^n$ can be found using the following power rule:

where:

• $\frac{d}{dx}$ represents the derivative with respect to $x$.
• $a$ is any rational constant.
• $n$ is any positive integer or zero.

Here are some key points to remember about the power rule:

• The derivative of $x^n$ is $nx^{(n-1)}$, where $n$ is any positive integer except for 1.
• The derivative of $x^1$ (i.e.,$x$) is simply 1.
• The derivative of a constant (i.e., where $n = 0$) is $0$.

Worked Examples

Let's use the power rule to find the derivatives of some functions:

Example 1:

Find the derivative of $f(x) = 3x^2$.

Solution:

Example 2:

Find the derivative of $g(x) = -2x^3$.

Solution:

Example 3:

Find the derivative of $h(x) = 5$.

Solution:

These examples demonstrate how the power rule can be applied to find the derivatives of various functions of the form $ax^n$.

Applying Derivatives

Once we know the derivative of a function, we can use it for various purposes, including:

• Finding the slope of the tangent line: The derivative of a function at a specific point ($x = a$) gives us the slope of the tangent line to the function's graph at that point.
• Identifying stationary points: A stationary point is a point where the derivative of the function is equal to zero ($f'(x) = 0$). These points may correspond to maxima, minima, or points of inflection.
• Analysing the behaviour of the function: By looking at the sign of the derivative (positive, negative, or zero), we can analyse whether the function is increasing, decreasing, or constant in different intervals.

Example

Find the derivative of the following function and state the intervals where the function is increasing and decreasing:

Solution:

1. Find the derivative:

2. Analyse the derivative:

$f'(x) = 0$ for $x = 0, 1$. Since $f'(x)$ is a polynomial, it is defined for all real numbers. Therefore, our points of interest are $x = 0$ and $x = 1$.

3. Sign chart:

We can create a sign chart to analyse the intervals where $f'(x)$ is positive/negative and, consequently, where the function is increasing/decreasing.

Explanation:

• The derivative is negative for x < 0, so the function is decreasing in this interval.
• The derivative is positive for 0 < x < 1, so the function is increasing in this interval.
• The derivative is negative for x > 1, so the function is decreasing in this interval.