Derivatives and Antiderivatives to Remember

f’(x)

f’(x)

1

f’(sinx)

cosx

f’(cosx)

-sinx

f’(tanx)

sec^2x

f’(secx)

secxtanx

f’(cscx)

-cscxcotx

f’(cotx)

-csc^2x

f’(a^x)

a^xlna

f’(e^x)

e^x

f’(lnx)

1/x

f’(loga(x))

1/(xlna)

f’(c)

0

f’(x^n)

nx^n-1

f’(f(g(x)))

f’(g(x)) * g’(x)

(f^-1)’(x)

1/(f’(f^-1(x))

f’(sec^-1(x))

1/( |x| sqrt(x^2 -1))

f’(csc^-1(x))

- 1/( |x| sqrt(x^2 -1))

f’(sin^-1(x))

1/ (sqrt(1- x^2))

f’(cos^-1(x))

- 1/ (sqrt(1- x^2))

f’(tan^-1(x))

1 / (1+ x^2)

f’(cot^-1(x))

- 1 / (1+ x^2)

(d/dx) (f(x)g(x))

f’(x)g(x) + f(x)g’(x)

(d/dx) (f(x)/g(x))

f’(x)g(x) - f(x)g’(x) / g(x)^2

ln 1

0

e^0

1

∫du

u + C

∫u^n (du)

(u^n+1)/n+1 + C

∫ 1/u (du)

ln |u| +C

∫ a^u (du)

a^u (1/ ln a) + C

∫ u dv

uv - ∫v du (Log Inverse trig Poly Trig Exp)

Fundamental Theory of Calculas

(d/dx) ∫ f(x) dx = f(x)

∫ f(x) dx = F(b) - F(a) where F’(x) = f(x)

2nd FTOC

(d/dx) ∫ f(x) dx = f(g(x)) g’(x)

Mean Value Theorem

if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in the interval (a,b) such that f'(c) is equal to the function's average rate of change over [a,b]: avg ROC = instant ROC at some point

Intermediate value theorem

if f(x) is continuous and f(a)<N<f(b), f ( a ) < N < f ( b ) , the line y=N intersects the function at some point x=c. Such a number c is between a and b and has the property that f(c)=N f ( c ) = N

Extreme Value Theorem

if a function is continuous on a closed interval [a,b], then the function must have a maximum and a minimum on the interva

∫ k f(x) dx

k ∫ f(x) dx

∫ [f(x) + g(x)]dx

∫ f(x) dx + ∫ g(x) dx

∫ (a to b) f(x) dx

- ∫ (b to a) f(x) dx

∫ (a to c) f(x) dx

∫ (a to b) f(x) dx + ∫ (b to c) f(x) dx

∫ (a to a) f(x) dx

0

f is integrable when

f is continuous over [a,b]

f is bounded on closed interval [a,b] and has at most a finite number of discontinuities

selecting techniques

• long division when degree of deno <= degree of num

• completing the square when degree of deno > degree of num

• partial fractions when degree of deno > degree of num that polynomials in deno can be factored into linear non-repeating

• u-sub

• integration by parts

• improper integrals: infinite interval or unbounded integral, can converge or diverge by replacing inifite part with variable

ln |0|

undefined

e ^x

exponential