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Back to Differential Calculus

Derivative Rules

Three rules do almost all the work. Each one comes from a picture: a tangent, a growing rectangle, and an interval stretched twice.

Power rule · step 1 of 19Speed

Three rules turn almost any derivative into arithmetic. First the power rule: the slope of xⁿ at x is n·xⁿ⁻¹. The green line is the tangent to x² at x = 1.

\frac{d}{dx}x^{2} = 2\,x^{1} \;\Rightarrow\; 2\,(1)^{1} = 2

\text{slope of } x^{2} \text{ at } x = 1 \text{ is } 2

Bring the power down as a factor, then lower it by one. It works for fractional and negative powers too: √x = x^½ and 1/x = x⁻¹.

Derivative Rules

Three rules do almost all the work. Each one comes from a picture: a tangent, a growing rectangle, and an interval stretched twice.

Power rule · step 1 of 19Speed

Three rules turn almost any derivative into arithmetic. First the power rule: the slope of xⁿ at x is n·xⁿ⁻¹. The green line is the tangent to x² at x = 1.

\frac{d}{dx}x^{2} = 2\,x^{1} \;\Rightarrow\; 2\,(1)^{1} = 2

\text{slope of } x^{2} \text{ at } x = 1 \text{ is } 2

Bring the power down as a factor, then lower it by one. It works for fractional and negative powers too: √x = x^½ and 1/x = x⁻¹.

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Derivative Rules

See also

Derivative Rules

See also