Graph Identity + Derivative Rules 2**

\frac{d}{dx}\ln x

\frac{1}{x}

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

\frac{1}{\sqrt{1-x^2}},x\ne\pm1

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

\frac{-1}{\sqrt{1-x^2}},x\ne\pm1

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

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

\frac{d}{dx}\left(\cot^{-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},x\ne\pm1,0

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

\frac{-1}{\left|x\right|\sqrt{x^2}-1},x\ne\pm1,0

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

\ln\left(a\right)a^{x}

\frac{d}{dx}\log_{a}x=

\frac{1}{x\ln a}

Power Rule

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

Product Rule

\frac{d}{dx}\left(uv\right)=u^{\prime}v+uv^{\prime}

Quotient Rule

\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u^{\prime}v-uv^{\prime}}{v^2}

Chain Rule

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

Definition of a Derivative

f^{\prime}\left(a\right)=\lim_{h\to0}\frac{f\left(a+h\right)-f\left(a\right)}{h}

Alternate Definition of a Derivative

f^{\prime}\left(a\right)=\lim_{x\to a}\frac{f\left(x\right)-f\left(a\right)}{x-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(\csc x\right)=

-\csc x\cdot\cos x

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

-\csc^2x

y=e^x

y=e^x+5

y=e^-x

y=-e^x

y=-e^-x

y=lnx

y=ln(x+5)

y=ln(-x)

y=-lnx

y=-ln(-x)

constant rule

derivative of a constant is zero.

constant multiple rule

derivative of a constant multiplied by a fxn is the constant times the derivative of the fxn.

power rule

the derivative of x^n is n*x^(n-1), where n is a real number.

sum and difference rule

the derivative of a sum or difference of fxns is the sum or difference of their derivatives.

product rule

the derivative of a product of two fxns is the first fxn times the derivative of the second plus the second fxn times the derivative of the first.

quotient rule

derivative of a quotient of two fxns is the denom x the derivative of the num - the num x the derivative of the denom, / denom²

chain rule

differentiate the outer fxn while keeping the inner fxn the same. multiply this by the derivative of the inner function.

derivative of e^x

e^x itself

derivative of a^x

a^x ln(a)

derivative of e^g(x)

e^g(x) g'(x)

derivative of a^g(x)

ln(a) a^g(x) g’(x)

derivative of ln(x)

1/x, x>0

derivative of ln(g(x))

g'(x)/g(x)

derivative of log_a(x)

1/xln(a), x>0

derivative of log_ag(x)

g’(x)/g(x)ln(a)

derivative of sin x

cos x

derivative of cos x

-sin x

derivative of tan x

sec² x

derivative of csc x

-csc x cot x

derivative of sec x

sec x tan x

derivative of cot x

-csc² x

half angle of sin² x

1/2(1-cos(2x)) - sin s for sad, sad face :( minus

half angle of cos² x

1/2(1+cos(2x)) - cos c for congratulations, happy face :) plus

integral power rule

add one to the exponent and bring 1/n+1 to the front

integral constant multiples rule

any constant in an integral is brought to the front

integral of exponential e

integral of exponential a

integral sums and differences

arcsin

Domain: [-1, 1]

Range: [-π/2, π/2]

arccos

Domain: [-1, 1]

Range: [0, π]

arctan

Domain: [-∞, ∞]

Range: [-π/2, π/2]

arccot x

D: (-∞,∞)

R: (0,π)

arcsec x

D: (⁻∞ , -1] U [1 , + ∞)

R: [0 , π/2) U (π/2 , π]

arccsc x

D: (⁻∞ , -1] U [1 , + ∞)

R: (-π/2, 0) U (0, π/2]

sin(x)

cos(x)

tan(x)

sec(x)

cosec(x)

cot(x)

y = x⁻²

Domain: (-ꝏ,0) U (0,ꝏ)

Range: (0,ꝏ)

y = x²

Domain: (-ꝏ,ꝏ)

Range: [0,ꝏ)

y = x³

Domain: (-ꝏ,ꝏ)

Range: (-ꝏ,ꝏ)

y = x¹/²

Domain: [0, ꝏ)

Range: [0,ꝏ)

y = x¹/³

Domain: (-ꝏ,ꝏ)

Range: (-ꝏ,ꝏ)

y = x⁻¹ or 1/x

Domain: (-ꝏ,0) U (0,ꝏ)

Range: (-ꝏ,0) U (0,ꝏ)

y = sin(x)

Domain: (-ꝏ,ꝏ)

Range: [-1,1]

y = cos(x)

Domain: (-ꝏ,ꝏ)

Range: [-1,1]

y = tan(x)

Domain: {x ≠ (2k+1)π/2}

Range: (-ꝏ,ꝏ)

y = cot(x)

Domain: {x ≠ kπ}

Range: (-ꝏ,ꝏ)

y = sec(x)

Domain: {x ≠ (2k+1)π/2}

Range: (-ꝏ,-1] U [1,ꝏ)

y = csc(x)

Domain: {x ≠ kπ}

Range: (-ꝏ,-1] U [1,ꝏ)

y = sin⁻¹(x)

Domain: [-1,1]

Range: [-π/2,π/2]

y = cos⁻¹(x)

Domain: [-1,1]

Range: [0,π]

y = tan⁻¹(x)

Domain: (-ꝏ,ꝏ)

Range: (-π/2,π/2)

y = cot⁻¹(x)

Domain: (-ꝏ,ꝏ)

Range: (0,π)

y = sec⁻¹(x)

Domain: (-ꝏ,-1] U [1,ꝏ)

Range: [0,π/2) U (π,3π/2)

y = csc⁻¹(x)

Domain: (-ꝏ,-1] U [1,ꝏ)

Range: (0,π/2) U (π,3π/2)

f(x) = bˣ, if b>1

Domain: (-ꝏ,ꝏ)

Range: (0,ꝏ)

f(x) = bˣ, if 0 < b < 1

Domain: (-ꝏ,ꝏ)

Range: (0,ꝏ)

f(x) = log_b(x), b > 1

Domain: (0,ꝏ)

Range: (0,ꝏ)

f(x) = logb(x), 0 < b < 1

Domain: (0,ꝏ)

Range: (0,ꝏ)

Linear function

f(x)= Mx + b

Quadratic / Square function

F(x) = x²

Cubic function

F(x)= x³

Square root function

F(x) = √x

Cube root function

F(x) = 3√x

Reciprocal / Rational function (Odd)

F(x)= 1/x or f(x) = 1/x^n → n is an odd number

Absolute value function

F(x) = | x |