Math 241 Calc I: Derivative Rules

\frac{d}{dx}\left(c\right)

0

\frac{d}{dx}\left(kx\right)

k

\frac{d}{dx}\left(x^{n}\right)

nx^{n-1}

\frac{d}{dx}\left(e^{x}\right)

e^{x}

\frac{d}{dx}\left(a^{x}\right)

a^{x}\ln a

\frac{d}{dx}\left(\ln x\right)

\frac{1}{x}

\frac{d}{dx}\left(\log_{a}x\right)

\frac{1}{x\ln a}

\frac{d}{dx}\left(\sin x\right)

\cos x

\frac{d}{dx}\left(\cos x\right)

-\sin x

\frac{d}{dx}\left(\tan x\right)

\sec^2x

\frac{d}{dx}\left(\sec x\right)

\sec x\tan x

\frac{d}{dx}\left(\cot x\right)

-\csc^2x

\frac{d}{dx}\left(\csc x\right)

-\csc x\cot x

\frac{d}{dx}\left(\sin^{-1}x\right)

\frac{1}{\sqrt{1-x^2}}

\frac{d}{dx}\left(\cos^{-1}x\right)

-\frac{1}{\sqrt{1-x^2}}

\frac{d}{dx}\left(\tan^{-1}x\right)

\frac{1}{1+x^2}

\frac{d}{dx}\left(\sec^{-1}x\right)

\frac{1}{\left|x\right|\sqrt{x^2-1}}

\frac{d}{dx}\left(\cot^{-1}x\right)

-\frac{1}{1+x^2}

\frac{d}{dx}\left(\csc^{-1}x\right)

-\frac{1}{\left|x\right|\sqrt{x^2-1}}

\frac{d}{dx}\left(cf\left(x\right)\right)

c\cdot f^{\prime}\left(x\right)

\frac{d}{dx}\left(f\left(x\right)+g\left(x\right)\right)

f^{\prime}\left(x\right)+g^{\prime}\left(x\right)

\frac{d}{dx}\left(f\left(x\right)\cdot g\left(x\right)\right)

f^{\prime}\left(x\right)g\left(x\right)+f\left(x\right)g^{\prime}\left(x\right)

\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)

\frac{f^{\prime}\left(x\right)g\left(x\right)-f\left(x\right)g^{\prime}\left(x\right)}{\left(g^{\prime}\left(x\right)\right)^2}

Limit definition of a derivative

f^{\prime}\left(a\right)=\lim_{h\to o}\frac{f\left(a+h\right)-f\left(a\right)}{h}  at x=a

secant line

a line passing through two points of a curve

tangent line

a line that touches a curve at a single point

derivative

instantaneous rate of change; slope of tangent

implicit differentiation

a technique for computing dy/dx for a function defined by an equation

\int_{}^{}\!k\,dx

0

\int_{}^{}\!x^{n}\,dx

\frac{\left(x^{n+1}\right)}{n+1}

\int_{}^{}\!\frac{1}{x}\,dx

ln|x|

\int_{}^{}\!e^{x}\,dx

e^x

\int_{}^{}\!\cos x\,dx

sinx

\int_{}^{}\!\sin x\,dx

-cosx

\int_{}^{}\!\sec^2\left(x\right)dx

tan(x)

\int_{}^{}\!\sec x\tan x\,dx

sec(x)

\int_{}^{}\!\csc^2x\,dx

-cotx

\int_{}^{}\!\csc x\cot x\,dx

-cscx

\int_{}^{}\!\frac{1}{x^2+1}\,dx

tan^-1(x)

\int_{}^{}\!\frac{1}{\sqrt{1-x^2}}\,dx

sin^-1(x)

\int_{}^{}\!\frac{1}{x\sqrt{x^2-1}}\,dx

sec^-1(x)

\int_{}^{}\!a^{x}\,dx

a^x / ln a

\int_{}^{}\!\tan x\,dx

ln|sec x|

\int_{}^{}\cot x\,dx

ln|sin x|

\int_{}^{}\!\sec x\,dx

ln|sec x + tan x|

\int_{}^{}\csc x\,dx

ln|csc x - cot x|

Avg value of a function

\frac{1}{b-a}\int_{}^{}f\left(x\right)\,dx

Trapezoidal rule

(interval / 2) initial area + 2 times area + final area

Simpson’s Rule

(interval / 3) initial area + alternate 4 and 2 times area + final area

\frac{d}{dx}\left(f\left(g\left(x\right)\right)\right)

f’(g(x)) * g’(x)