# Pre-Calculus/Calculus: Reference Sheet

This page is a reference sheet which contains key calculus differentiation and integration formulas.  Be sure to check the formula database for other formulas.

Differentiation Formulas
Integration Formulas

## Differentiation Formulas

• d/dx c = 0, c constant
• d/dx cf(x) = cf'(x), c constant
• d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
• d/dx [f(x) * g(x)] = f(x) * g'(x) + g(x) * f'(x)    (product rule)
• d/dx [f(x) / g(x)] = (g(x)f'(x) - f(x)g'(x))/([g(x)]^2)  (quotient rule)
• d/dx f[g(x)] = f'[g(x)] * g'(x)
OR
for u = g(x), d/dx f(u) = f'(u) * u' = f'(u) * g'(x)
OR
dy/dx = dy/du * du/dx        (these are all chain rule)
 GENERAL SPECIFIC d/dx u^n = nu^(n-1) * u' d/dx ln u = u'/u d/dx e^u = e^u * u' d/dx sin u = cos u * u' d/dx cos u = -sin u * u' d/dx tan u = sec^2 u * u' d/dx arcsin u = u'/(SQRT(1 - u^2)) d/dx arctan u = u'/(1 + u^2) d/dx cot u = -csc^2 u * u' d/dx sec u = sec u tan u * u' d/dx csc u = -csc u cot u * u' d/dx a^u * u' ln a d/dx log(SUB a)u = u'/(u ln a) d/dx x^n = nx^(n - 1) d/dx ln x = 1/x d/dx e^x = e^x d/dx sin x = cos x d/dx cos x = -sin x d/dx tan x = sec^2 d/dx arcsin x = 1/(SQRT(1 - x^2)) d/dx arctan x = 1/(1 + x^2) d/dx cot x = -csc^2 x d/dx sec x = sec x tan x d/dx csc x = -csc x cot x d/dx a^x = a^x ln a d/dx log(SUB a)x = 1/(x ln a)

## Integration

Remember that u'dx = du and may be replaced by du.

(On this page, a capital S will be used as an integration symbol because text only browsers do not display the integration symbol.)

• Sdx = x + c
• Sk f(x) dx = k Sf(x) dx
• S[f(x) ± g(x)]dx = Sf(x)dx ± Sg(x)dx
 GENERAL SPECIFIC Su^n u' dx = ((u^(n + 1)/(n + 1)) + c S(u'/u)dx = ln |u| + c Se^u u' dx = e^u + c Scos u * u'dx = sin u + c Ssin u * u'dx = -cos u + c Ssec^2 u * u'dx = tan u + c Stan u * u'dx = -ln |cos u| + c OR ln |sec u| + c Scot u * u'dx = ln |sin u| + c Sa^u * u'dx = ((a^u)/(ln a)) + c Ssec u * tan u * u'dx = sec u + c Scsc^2 u * u'dx = -cot u + c Scsc u cot u * u'dx = -csc u + c S(u'/(a^2 + u^2))dx = (1/a)arctan(u/a) + c S(u'/(SQRT(a^2 - u^2)))dx = arcsin(u/a) + c S(a^u)u' dx = ((a^u)/(ln a)) + c Sx^n dx = ((x^(n + 1))/(n + 1)) + c S(1/x)dx OR S(dx/x) = ln |x| + c Se^x dx = e^x + c S(cos x)dx = sin x + c S(sin x)dx = -cos x + c S(sec^2 x)dx = tan x + c S(tan x)dx = -ln |cos x| + c OR ln |sec x| + c S(cot x)dx = ln |sin x| + c Sa^x dx = ((a^x)/(ln a)) + c Ssec x tan x dx = sec x + c S(csc^2 x)dx = -cot x + c Scsc x cot x dx = -csc x Sdx/(a^2 + x^2) dx = 1/a arctan (x/a) + c Sdx/(SQRT(a^2 - x^2)) dx = arcsin (x/a) + c Sa^x dx = ((a^x)/(ln a)) + c

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Math for Morons Like Us -- Pre-Calculus/Calculus: Reference Sheet
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