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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 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 un = nun-1 * u' d/dx lnu = u'/u d/dx eu = eu * u' d/dx sinu = cosu * u' d/dx cosu = -sinu * u' d/dx tanu = sec2u * u' d/dx arcsinu = u'/(SQRT(1 - u2)) d/dx arctanu = u'/(1 + u2) d/dx cotu = -csc2u * u' d/dx secu = secu tanu * u' d/dx cscu = -cscu cotu * u' d/dx au * u' ln a d/dx logau = u'/(u ln a) d/dx xn = nxn - 1 d/dx lnx = 1/x d/dx ex = ex d/dx sinx = cosx d/dx cosx = -sinx d/dx tanx = sec2 d/dx arcsinx = 1/(SQRT(1 - x2)) d/dx arctanx = 1/(1 + x2) d/dx cotx = -csc2x d/dx secx = secx tanx d/dx cscx = -cscx cotx d/dx ax = ax ln a d/dx logax = 1/(x ln a) Remember that u'dx = du and may be replaced by du. dx = x + c k f(x) dx = k f(x) dx [f(x) ± g(x)]dx = f(x)dx ± g(x)dx GENERAL SPECIFIC un u' dx = ((un + 1)/(n + 1)) + c (u'/u)dx = ln|u| + c eu u' dx = eu + c cosu * u'dx = sinu + c sinu * u'dx = -cosu + c sec2u * u'dx = tanu + c tanu * u'dx = -ln |cosu| + c or ln |secu| + c cotu * u'dx = ln |sinu| + c au * u'dx = ((au)/(lna)) + c secu * tanu * u'dx = secu + c csc2u * u'dx = -cotu + c cscu cot u * u' dx = -cscu + c (u'/(a2 + u2))dx = (1/a)arctan(u/a) + c (u'/(SQRT(a2 - u2)))dx = arcsin(u/a) + c auu' dx = ((au)/(ln a)) + c xn dx = ((xn + 1)/(n + 1)) + c (1/x)dx OR (dx/x) = ln|x| + c exdx = ex + c cosx dx = sin x + c sinx dx = -cos x + c sec2x dx = tan x + c tanx dx = -ln |cosx| + c or ln |secx| + c cotx dx = ln |sinx| + c ax dx = ((ax)/(ln a)) + c secx tanx dx = sec x + c csc2x dx - -cotx + c cscx cotx dx = -csc x dx/(a2 + x2) dx = 1/a arctan (x/a) + c dx/(SQRT(a2 - x2))dx = arcsin (x/a) + c ax dx = ((ax)/(ln a)) + c