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1. Basic Derivative formulae (xn )0 = nxn−1 (ax )0 = ax ln a (loga x)0 = (ex )0 = ex 1 x ln a (ln x)0 = 1 x (sin x)0 = cos x (cos x)0 = − sin x (tan x)0 = sec2 x (cot x)0 = − csc2 x (sec x)0 = sec x tan x (csc x)0 = − csc x cot x 1 1 − x2 1 (tan−1 x)0 = 1 + x2 1 (sec−1 x)0 = √ x x2 − 1 (cos−1 x)0 = √ −1 1 − x2 −1 (cot−1 x)0 = 1 + x2 −1 −1 0 (csc x) = √ x x2 − 1 (sin−1 x)0 = √ 2. Differentiation Rules Sum rule: (f + g)0 = f 0 + g 0 wheref = f (x), g = g(x) Product rule: (f · g)0 = f 0 g + g 0 f 0 f 0 g − g0 f f = Quotient rule: g g2 0 Chain rule: [f (g)] = f 0 (g) · g 0 or dy = dx dy du du · dx Implicit differentiation: If y = y(x) is given implicitly, find derivative to the entire equation with respect to x. Then solve for y 0 . 3. Identities of Trigonometric Functions sin x cos x 1 sec x = cos x tan x = sin2 x + cos2 x = 1 cos x sin x 1 csc x = sin x cot x = 1 + tan2 x = sec2 x 1 + cot2 x = csc2 x 4. Laws of Exponential Functions and Logarithms Functions ax · ay = ex+y loga (xy) = loga (x) + loga (y) ax = ax−y ay loga ( xy ) = loga (x) − loga (y) (ax )y = axy loga (xn ) = n loga (x) aloga (x) = x ln x = loge x a0 = 1 loga a = 1, loga 1 = 0 1