The Knowledge Web

If you take the derivative of $x^2$ , you get $2x$ . If you take the derivative of $x^3$ , you get $3x^2$ . Do you see the pattern?

\frac{d}{dx}(x^n) = nx^{n-1}

The rule is simple: Bring the power down, then subtract one from the exponent.

Polynomial Analyzer

Notice how the derivative is always exactly one degree lower than the original function.

Original

x²

Derivative

2x¹

Toggle the power to see how the derivative function (Red) relates to the original (Blue).

What if you have multiple terms added together? $f(x) = x^3 + x^2$ .

Good news: The derivative is a linear operator. This means you can differentiate each piece separately and just add them up.

y = x^5 + x^3

y' = 5x^4 + 3x^2

y = 3x^2 + 10x

y' = 6x + 10