CALC STUFF -- DERIVATIVE AND INTEGRAL RULES

d/dx [x^n] = nx^n-1

power rule

d/dx = [kx] = k

constant

d/dx [uv] = u’v +uv’

product rule

d/dx [u/v] = (u’v - uv’)/ v²

quotient rule

d/dx [sinx] = cosx

sine

d/dx [cosx] = -sinx

cosine

d/dx [tanx] = sec²x

tangent

d/dx [cotx] = -csc²x

cotangent

d/dx [secx] = tanxsecx

secant

d/dx [cscx] = -cscxcotx

cosecant

d/dx [f(g(x)] = f’(g(x)) * g’(x)

chain rule

d/dx [sin^-1x] = 1/√(1-x²)

inverse sine

d/dx [cos^-1] = - 1/√(1-x²)

inverse cosine

d/dx [tan^-1] = 1/1+x²

inverse tangent

d/dx [cot^-1] = -1/1+x²

inverse cot

d/dx [sec^-1] = 1/|x| √(x²-1)

inverse secant

d/dx [csc^-1] = -1/|x| √(x²-1)

inverse cosecant

d/dx [e^x]= e^x

exponential

d/dx [lnx] = 1/x

natural log

d/dx [a^x] = a^x * ln(a)

constant to a power

d/dx [log(basea) x] = 1/xlna

logarithmic function

d/dx [f^-1(x)] = 1/f’(f^-1(x))

inverse functions

∫f(x) dx [a,a] = 0

zero rule

∫f(x) dx [a,b] + ∫f(x) dx [b,c] = ∫f(x) dx [a,c]

additivity

∫f(x) dx [b,a] = - ∫f(x) dx [a,b]

order of integration

∫kf(x) dx [a,b] = k ∫f(x) dx [a,b]

constant multiple (divide out constant)

∫(f(x) ± g(x)) dx [a,b] = ∫f(x) dx [a,b] ± ∫fg(x) dx [a,b]

sum/ difference

f(x) >/= g(x) on [a.b] → ∫f(x) dx [a,b] >/= ∫g(x) dx [a,b]

f(x) >/= 0 on [a.b] → ∫f(x) dx [a,b] >/= 0

domination

a(f) = 1/ b-a ∫f(x) dx [a,b]

average mean value theorem

∫f(t) dt [a,x] = F(x) - F(a)

fundamental theorem of calculus

d/dx ∫f(t) dt [a,x] = f(x)

derivative of an antiderivative