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:

$f'(x) = \frac{d}{dx}(3x^2) = (2)(3)x^{(2-1)} = 6x$**Example 2:**

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

#### Solution:

$g'(x) = \frac{d}{dx}(-2x^3) = (-2)(3)x^{(3-1)} = -6x^2$**Example 3:**

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

#### Solution:

$h'(x) = \frac{d}{dx}(5) = (0)(5)x^{(0-1)} = 0$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:

$f(x) = 2x^3 - 3x^2 + 5$**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.