501DerivPoly

Derivative of a Polynomial

For any term in a polynomial: math \dfrac {d}{dx} \left( ax^n \right) = nax^{n-1} math

Example math y=x^3+4x^2-3x+2 math The term, –3x, can be written as –3x 1. The constant term, +2, can be written as +2x 0.

math y=x^3+4x^2-3x^1+2x^0 math

The derivative is: math \\ \dfrac{dy}{dx}=3x^2+8x^1-3x^0+0 \\ \\ \Rightarrow \; \; =3x^2+8x-3 math

This rule works when n is negative and when n is a fraction. math \textbf{Eg: } y = x^{-2} = \dfrac {1}{x^2} math math \Rightarrow \dfrac {dy}{dx} = -2x^{-3} = \dfrac {-2}{x^3} math

math \textbf{Eg } y = x^{\frac{1}{2}} = \sqrt{x} math math \Rightarrow \dfrac {dy}{dx} = \dfrac {1}{2}x^{-\frac{1}{2}} = \dfrac{1}{2\sqrt{x}} math

**See also** [|www.mathsonline.com.au] Y12Advanced --> Calculus --> Introductory Calculus --> Lesson 3