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:
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: \[\text{if} \quad y=ax^n \quad \text{then} \quad \frac{dy}{dx}=n\times ax^{n-1}\]
Differentiate each of the following:
Using the power rule for differentiation:
Consider the curve defined by: \[y=2x^2\]
Differentiate each of the following:
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:
\[f(x)=a.\sqrt[n]{x^m} \quad \text{and} \quad f(x)=\frac{a}{\sqrt[n]{x^m}}\]
Differentiate each of the following:
Each of these can be differentiated using the power rule.