Power rule, constant multiples, sums and differences, with practice problems

Q: What is the derivative of a constant?

0. A constant is a multiple of x^0, and bringing the exponent 0 down gives 0 times something.

Q: What is the derivative of x?

1. Bring down the exponent 1 of x^1, and x^0=1 remains.

\begin{gathered} (x^n)'=nx^{n-1}, \\[4pt] (kf)'=kf', \\[4pt] (f\pm g)'=f'\pm g' \end{gathered}

For a power of $x$ , bring the exponent down in front and reduce the exponent by 1. Constant multiples come straight out, and sums and differences are differentiated term by term.

(3x^4-5x+7)'=12x^3-5

These three rules cover every polynomial. A constant term differentiates to $0$ .

Examples

01Differentiate $\displaystyle \frac{d}{dx} x^{4}$ AnswerHide

Answer $\displaystyle 4x^{3}$
02Differentiate $\displaystyle \frac{d}{dx} x^{2}$ AnswerHide

Answer $\displaystyle 2x$
03Differentiate $\displaystyle \frac{d}{dx} x^{3}$ AnswerHide

Answer $\displaystyle 3x^{2}$
04Differentiate $\displaystyle \frac{d}{dx} x$ AnswerHide

Answer $\displaystyle 1$
05Differentiate $\displaystyle \frac{d}{dx} x^{5}$ AnswerHide

Answer $\displaystyle 5x^{4}$