Jorge Valle

Jorge Valle

Jorge Valle is a front end developer with a particular passion for, and expertise in, JavaScript and user interfaces. Lately, he's also been diving into machine learning.

The power rule in differential calculus

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.

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