20 min read

The Power Rule

A shortcut for differentiating any power of x, without repeating the full limit process every time.

The Shortcut

A constant has derivative 0 — its graph is a flat line, with zero gradient everywhere.

Example 1 — a higher power

Differentiate f(x) = x^{5} . 1 Bring the power (5) down in front: 5 x ... 2 Reduce the power by 1 : 5 - 1 = 4 Answer: f'(x) = 5 x^{4}

Example 2 — x on its own

Differentiate f(x) = x . 1 Think of x as x^{1} . Bring the power (1) down: 1 x ... 2 Reduce the power by 1 : 1 - 1 = 0, and x^{0} = 1 Answer: f'(x) = 1

Practice

What is the derivative of x^{7} ?

20 min read

A shortcut for differentiating any power of x, without repeating the full limit process every time.

A constant has derivative 0 — its graph is a flat line, with zero gradient everywhere.

Example 1 — a higher power

Differentiate f(x) = x^{5} . 1 Bring the power (5) down in front: 5 x ... 2 Reduce the power by 1 : 5 - 1 = 4 Answer: f'(x) = 5 x^{4}

Example 2 — x on its own

Differentiate f(x) = x . 1 Think of x as x^{1} . Bring the power (1) down: 1 x ... 2 Reduce the power by 1 : 1 - 1 = 0, and x^{0} = 1 Answer: f'(x) = 1

Practice

What is the derivative of x^{7} ?