The power rule

The power rule is very simple and easy to remember. It dictates how to differentiate functions of the form $ f(x) = x^n $ .

$$ f(x) = x^n \\ f'(x) = nx^{n-1} $$
Figure 1: The power rule defined.

Here's a simple example.

$$ f(x) = x^5 \\ \begin{align} f'(x) & = 5x^{5-1} \\ & = 5x^4 \end{align} $$
Figure 2: Differentiating using the power rule.

What happens if there is a coefficient present? The same pattern is retained. We apply the power rule to the variable term, and then multiply by the coefficient.

$$ f(x) = 2x^3 \\ \begin{align} f'(x) & = 2 \times 3x^{3-1} \\ & = 2 \times 3x^{2} \\ & = 6x^2 \end{align} $$
Figure 3: What happens if there's a coefficient present?

The rule is also applicable for negative powers.

$$ f(x) = 2x^{-3} \\ \begin{align} f'(x) & = 2 \times -3x^{-3-1} \\ & = 2 \times -3x^{-4} \\ & = -6x^{-4} \end{align} $$
Figure 4: Example with negative powers.

Differentiating a polynomial

We can use the power rule when differentiating polynomials as well. We apply the rule to each term individually, and keep the same structure.

$$ f(x) = 3x^3 + 4x^5 \\ \begin{align} f'(x) & = (3 \times 3x^{3-1}) + (4 \times 5x^{5-1}) \\ & = (3 \times 3x^{2}) + (4 \times 5x^{4}) \\ & = 9x^2 + 20x^4 \end{align} $$
Figure 5: Polynomials can also be differentiated via the power rule.