Trig Derivatives

f(x) = sin x
f’(x) =

cos x

f(x) = cos x
f’(x) =

−sin x

f(x) = tan x

f’(x) =

sec² x

f(x) = csc x

f’(x) =

−csc x cot x

f(x) = sec x
f’(x) =

sec x tan x

f(x) = cot x
f’(x) =

−csc² x

product rule

(fg)' = f’ g + g' f

quotient rule

(u/v)' = (v u' − u v') / v²

chain rule

dy/dx = dy/du · du/dx

power rule

d/dx(xⁿ) = n xⁿ⁻¹

constant multiple rule

d/dx(c·f(x)) = c·f'(x)

sum rule

d/dx(f(x) ± g(x)) = f'(x) ± g'(x)

exponential rule

d/dx(eˣ) = eˣ

natural log rule

d/dx(ln x) = 1/x

implicit differentiation

differentiate both sides with respect to x

related rates

differentiate with respect to time t

mean value theorem

f'(c) = [f(b) − f(a)] / (b − a)

fundamental theorem of calculus (part 1)

d/dx(∫ₐˣ f(t) dt) = f(x)

fundamental theorem of calculus (part 2)

∫ₐᵇ f(x) dx = F(b) − F(a)

first derivative test

sign of f' determines increasing/decreasing

second derivative test

sign of f'' determines concavity

critical points

f'(x) = 0 or undefined

area under a curve

∫ f(x) dx

average value of a function

(1/(b − a)) ∫ₐᵇ f(x) dx