Derivatives — Key Formulas

Summary:  To work efficiently in calculus, you need to memorize about a dozen formulas for derivatives. This page lists them.

You can work faster and more accurately in calculus if you have a lot of formulas ready to hand. Your textbook lists most of them on the inside front and back covers, but you’ll use a few formulas so often that looking them up will slow you down. Also, you need to be thoroughly familiar with the formulas or you may miss an easy method of solving a problem.

It’s worth your while to memorize the formulas listed below. If the symbols don’t seem memorable, try memorizing the formulas in words. For easy reference to examples and proofs, each formula is keyed to the page of your textbook where it’s presented.

Your textbook generally uses the notation; this sheet often uses the prime (′) notation because it’s easier to type and might be easier to memorize.

The Power Rule

(page 106) the power rule for any rational n: [xⁿ] = n x^n–1

(page 106) special case n=1, derivative of x: [x] = 1

(page 105) special case n=0, derivative of a constant: [c] = 0

Derivatives of Trig Functions

(page 110) sine and cosine — memorize these:

(sin x)′ = cos x
(cos x)′ = –sin x

(page 121) the other four functions — Either memorize these or work them out at need using the quotient rule and power rule, but at least be generally familiar with them.

Rules for Transforming Derivatives

u and v stand for functions; c and n stand for constants.

(page 108) constant multiple rule:

(cu)′ = c ′

(page 109) sum and difference rules:

(u ± v)′ = u′ ± v′

(page 117) product rule — very, very important:

(uv)′ = u′v + uv′

(page 119) quotient rule — also very important:

(u/v)′ = (u′v – uv′) / v²

(Caution: Don’t overuse the quotient rule. Simplify the fraction first, and then use an easier rule if possible.)

(page 128) chain rule — ultra super extra double important:

where y is the “outer” function of u and u is the “inner” function of x. The chain rule is used with all the other rules to differentiate things like sin²(6x).

Position, Velocity, Acceleration

(page 123) know which one is the rate of change of the other one:

v(t) = s′(t)
a(t) = v′(t) = s′′(t)