Power Rule for Differentiation

In this section we learn how to differentiate, find the derivative of, any power of $$x$$.
That's any function that can be written: $f(x)=ax^n$ We'll see that any function that can be written as a power of $$x$$ can be differentiated using the power rule for differentiation.
In particular we learn how to differentiate when:

• the power is a positive integer like $$f(x) = 3x^5$$.
• the power is a negative number, this means that the function will have a "simple" power of $$x$$ on the denominator like $$f(x) = \frac{2}{x^7}$$.
• the power is a fraction, this means that the function will have an $$x$$ under a root like $$f(x) = 5\sqrt{x}$$.
We start by learning the formula for the power rule.

Power Rule

Given a function which is a power of $$x$$, $$f(x)=ax^n$$, its derivative can be calculated with the power rule: $\text{if} \quad f(x)=ax^n \quad \text{then} \quad f'(x)=n\times ax^{n-1}$ We can also write this: $\text{if} \quad y=ax^n \quad \text{then} \quad \frac{dy}{dx}=n\times ax^{n-1}$

Tutorial 1: Power Rule for Differentiation

In the following tutorial we illustrate how the power rule can be used to find the derivative function (gradient function) of a function that can be written $$f(x)=ax^n$$, when $$n$$ is a positive integer.

Example 1

Find the derivative of the function defined by: $f(x) = 2x^4$

Detailed Solution

Comparing the function $$f(x) = 2x^4$$ to the generic "power function" $$f(x) = ax^n$$, we can see that: $a = 2 \quad \text{and} \quad n = 4$ The power rule for differentiation: $f'(x) = n\times ax^{n-1}$ therfore leads to: \begin{aligned} f'(x) & =4\times 2x^{3-1} f'(x) & = 8x^2 \end{aligned} The derivative is therefore: $f'(x) = 8x^2$

Exercise 1

Use the power rule for differentiation to find the derivative function of each of the following:

1. $$f(x) = 6x^3$$
2. $$y = x^4$$
3. $$f(x) = -2x^6$$
4. $$y = \frac{x^2}{2}$$
5. $$f(x) = 3x$$
6. $$y = \frac{2}{5}x^{10}$$
7. $$f(x) = -6x^3$$
8. $$y = 4x^4$$

1. For $$f(x) = 6x^3$$ we find: $f'(x) = 18x^2$
2. For $$y = x^4$$ we find: $\frac{dy}{dx} = 4x^3$
3. For $$f(x) = -2x^6$$ we find: $f'(x) = -12x^5$
4. For $$y = \frac{x^2}{2}$$ we find: $\frac{dy}{dx} = x$
5. For $$f(x) = 3x$$ we find: $f'(x) = 3$
6. For $$y = \frac{2}{5}x^{10}$$ we find: $\frac{dy}{dx} = 4x^9$
7. For $$f(x) = -6x^3$$ we find: $f'(x) = -18x^2$
8. For $$y = 4x^4$$ we find: $\frac{dy}{dx}= 16x^3$

Negative Exponents

The power rule also works for negative exponents.
Remember that: $\frac{a}{x^m} = ax^{-m}$ this allows us to use the power rule to differentiate any function that can be written: $f(x)=\frac{a}{x^m}$

Tutorial 2: Negative Exponents

In the following tutorial we illustrate how the power rule can be used to find the derivative function (gradient function) of a function that can be written $$f(x)=\frac{a}{x^m}$$, when $$m$$ is a positive integer.

We use the fact that $$\frac{a}{x^m} = a.x^{-m}$$ to then use the power rule.

Exercise 2

Differentiate each of the following:

1. $$f(x) = \frac{3}{x^2}$$
2. $$f(x) = \frac{1}{x}$$
3. $$y = \frac{5}{x^3}$$
4. $$f(x) = -\frac{5}{x^3}$$
5. $$y = \frac{6}{x^4}$$
6. $$f(x) = - \frac{2}{x}$$
7. $$y = \frac{3}{4x^2}$$
8. $$f(x) = -\frac{2}{3x^3}$$

Solution

1. For $$f(x) = \frac{3}{x^2}$$, we find: $f'(x) = -6.x^{-3}$ Which we can also write: $f'(x) = -\frac{6}{x^3}$
2. For $$f(x) = \frac{1}{x}$$ we find: $f'(x) = -1.x^{-2}$ Which can/should be written: $f'(x) = -\frac{1}{x^2}$
3. For $$y = \frac{5}{x^3}$$ we find: $\frac{dy}{dx} = -15.x^{-4}$ which can/should be written: $\frac{dy}{dx} = - \frac{15}{x^4}$
4. For $$f(x) = -\frac{5}{x^3}$$ we find: $f'(x) = 15.x^{-4}$ which can/should be written: $f'(x) = \frac{15}{x^4}$
5. For $$y = \frac{6}{x^4}$$ we find: $\frac{dy}{dx} = -24.x^{-4}$ which can/should be written: $\frac{dy}{dx} = -\frac{24}{x^5}$
6. For $$f(x) = - \frac{2}{x}$$ we find: $f'(x) = 2.x^{-2}$ which can/should be written: $f'(x) = \frac{2}{x^2}$
7. For $$y = \frac{3}{4x^2}$$ we find: $\frac{dy}{dx} = -\frac{3}{2}.x^{-3}$ which can/should be written: $\frac{dy}{dx} = -\frac{3}{2x^3}$
8. For $$f(x) = -\frac{2}{3x^3}$$ we find: $f'(x) = 2.x^{-4}$ which can/should be written: $f'(x) = \frac{2}{x^4}$

Fractional Exponents

The power rule for differentiation also works for any fraction.

Remembering that: $\sqrt[n]{x^m}=x^{\frac{m}{n}} \quad \text{and} \quad \frac{1}{\sqrt[n]{x^m}} = x^{-\frac{m}{n}}$ we can differentiate any function that can be written: $f(x)=a.\sqrt[n]{x^m} \quad \text{and} \quad f(x)=\frac{a}{\sqrt[n]{x^m}}$

Exercise 3

Differentiate each of the following:

1. $$f(x)=3.\sqrt{x}$$
2. $$y=5.\sqrt[3]{x^4}$$
3. $$y = 12.\sqrt[6]{x}$$
4. $$f(x) = 10.\sqrt[5]{x^3}$$
5. $$f(x) = -4.\sqrt{x^5}$$
6. $$y = 9.\sqrt[3]{x^2}$$
7. $$f(x)=\frac{4}{\sqrt{x}}$$
8. $$y = - \frac{8}{\sqrt{x^5}}$$

Solution

Each of these can be differentiated using the power rule.

1. For $$f(x)=3\sqrt{x}$$ we find: $f'(x) = \frac{3}{2}.x^{-\frac{1}{2}}$ which we can write: $f'(x) = \frac{3}{2\sqrt{x}}$
2. For $$y=5\sqrt[3]{x^4}$$ we find: $\frac{dy}{dx} = \frac{20}{3}x^{\frac{1}{3}}$ which we can write: $\frac{dy}{dx} = \frac{20}{3}\sqrt[3]{x}$
3. For $$f(x)=\frac{4}{\sqrt{x}}$$ we find: $f'(x) = -2.x^{-\frac{3}{2}}$ which we can write: $f'(x) = -\frac{2}{\sqrt{x^3} }$
4. For $$f(x) = 10.\sqrt[5]{x^3}$$ we find: $f'(x) = 6.x^{-\frac{2}{5}}$ which we can write: $f'(x) = \frac{6}{\sqrt[5]{x^2}}$
5. For $$f(x) = -4.\sqrt{x^5}$$ we find: $f'(x) = -10.x^{\frac{3}{2}}$ which we can also write: $f'(x) = - 10 \sqrt{x^3}$
6. For $$y = 9.\sqrt[3]{x^2}$$ we find: $\frac{dy}{dx} = 6.x^{-\frac{1}{3}}$ which we can also write: $\frac{dy}{dx} = \frac{6}{\sqrt[3]{x}}$
7. For $$f(x)=\frac{4}{\sqrt{x}}$$ we find: $f'(x) = -2.x^{-\frac{3}{2}}$ which we can also write: $f'(x) = - \frac{2}{\sqrt{x^3}}$
8. For $$y = - \frac{8}{\sqrt{x^5}}$$ we find: $\frac{dy}{dx} = 20.x^{-\frac{7}{2}}$ which we can also write: $\frac{dy}{dx} = \frac{20}{\sqrt{x^7}}$