CALCULUS BASICS

Derivative of a constant c

0

Power Rule: Derivative of x^n

n * x^(n-1)

Product Rule

f'(x)g(x) + f(x)g'(x)

Quotient Rule

[f'(x)g(x) - f(x)g'(x)] / [g(x)]^2

Chain Rule

f'(g(x)) * g'(x)

Derivative of e^x

e^x

Derivative of a^x

a^x * ln(a)

Derivative of ln(x)

1/x

Derivative of log_a(x)

1 / (x * ln(a))

Derivative of sin(x)

cos(x)

Derivative of cos(x)

-sin(x)

Derivative of tan(x)

sec^2(x)

Derivative of sec(x)

sec(x) * tan(x)

Derivative of csc(x)

-csc(x) * cot(x)

Derivative of cot(x)

-csc^2(x)

Derivative of arcsin(x)

1 / sqrt(1 - x^2)

Derivative of arccos(x)

-1 / sqrt(1 - x^2)

Derivative of arctan(x)

1 / (1 + x^2)

Derivative of arccot(x)

-1 / (1 + x^2)

Derivative of arcsec(x)

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

Derivative of arccsc(x)

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

Derivative of sinh(x)

cosh(x)

Derivative of cosh(x)

sinh(x)

Derivative of tanh(x)

sech^2(x)

Integral of x^n

[x^(n+1)] / (n+1) + C

Integral of 1/x

ln|x| + C

Integral of e^x

e^x + C

Integral of a^x

a^x / ln(a) + C

Integral of sin(x)

-cos(x) + C

Integral of cos(x)

sin(x) + C

Integral of sec^2(x)

tan(x) + C

Integral of csc^2(x)

-cot(x) + C

Integral of sec(x)tan(x)

sec(x) + C

Integral of csc(x)cot(x)

-csc(x) + C

Integral of tan(x)

ln|sec(x)| + C

Integral of cot(x)

ln|sin(x)| + C

Integral of sec(x)

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

Integral of csc(x)

-ln|csc(x) + cot(x)| + C

Integral of 1 / sqrt(a^2 - x^2)

arcsin(x/a) + C

Integral of 1 / (a^2 + x^2)

(1/a) * arctan(x/a) + C

Integral of 1 / (x * sqrt(x^2 - a^2))

(1/a) * arcsec(|x|/a) + C

Integral of sinh(x)

cosh(x) + C

Integral of cosh(x)

sinh(x) + C

Integration by Parts Formula

uv - integral of (v * du)

Integral of ln(x)

xln(x) - x + C

Taylor Series of f(x) at x=a

f(a) + f'(a)(x-a) + f''(a)(x-a)^2 / 2! + …

Maclaurin Series of f(x)

f(0) + f'(0)x + f''(0)x^2 / 2! + …

Maclaurin expansion of e^x

1 + x + (x^2)/2! + (x^3)/3! + …

Maclaurin expansion of sin(x)

x - (x^3)/3! + (x^5)/5! - …

Maclaurin expansion of cos(x)

1 - (x^2)/2! + (x^4)/4! - …

Maclaurin expansion of ln(1+x)

x - (x^2)/2 + (x^3)/3 - (x^4)/4 + …

Maclaurin expansion of 1/(1-x)

1 + x + x^2 + x^3 + …

Binomial Expansion of (1+x)^n

1 + nx + n(n-1)(x^2)/2! + …

Leibniz Rule for nth derivative of u*v

Sum from k=0 to n of [ nCk * u^(n-k) * v^(k) ]

Leibniz Integral Rule (Diff under integral)

f(x; b(x))b'(x) - f(x; a(x))a'(x) + integral of [partial derivative of f with respect to x]