Calculus Derv Review

Flashcards for calculus formulas and theorems.

d/dx sin(x)

cos(x)

d/dx cos(x)

-sin(x)

d/dx e^x

e^x

d/dx tan(x)

sec²(x)

d/dx cot(x)

-csc²(x)

d/dx sec(x)

sec(x)tan(x)

∫sin(u) du

-cos(u) + C

∫cos(u) du

sin(u) + C

∫e^x du

e^x + C

∫tan(u) du

-ln|cos(u)| + C

∫cot(u) du

ln|sin(u)| + C

∫sec(u) du

ln|sec(u) + tan(u)| + C

d/dx logₐ(u)

u'/(ln(a) * u)

d/dx ln|u|

u'/u

d/dx aᵘ

ln(a) * aᵘ * u'

d/dx eᵘ

eᵘ * u'

d/dx arcsin(u)

u'/√(1-u²)

d/dx arccos(u)

-u'/√(1-u²)

d/dx arctan(u)

u'/(1+u²)

∫sec²(u) du

tan(u) + C

∫csc²(u) du

-cot(u) + C

∫sec(u)tan(u) du

sec(u) + C

∫csc(u)cot(u) du

-csc(u) + C

∫du/u

ln|u| + C

∫aᵘ du

aᵘ/ln(a) + C

Intermediate Value Theorem (IVT)

If f(x) is continuous on [a,b], then there exists a 'c' on (a,b) where f(a) < f(c) < f(b) or f(a) > f(c) > f(b)

Extreme Value Theorem (EVT)

If f(x) is continuous on [a,b], then f(x) has both an absolute minimum and an absolute maximum on [a,b]. Use the Candidate's Test to solve.

Mean Value Theorem (MVT)

If f(x) is continuous on [a,b] and differentiable on (a,b), then there exists a 'c' in (a,b) such that f'(c) = (f(b) - f(a))/(b-a), where f'(c) is the Instantaneous Rate of Change (IROC) and (f(b) - f(a))/(b-a) is the Average Rate of Change (AROC).