Derivative and Integral Reference Guide Differentiation Rules Linearity Product & Quotient Rules Chain Rule 0d 0 u+v=u +v 0 0 d 0 0 dx uv = u v + v u dx f(u) = f (u) · u d dx 0d hu i d dx cu = cu dx 0 0 v =u v − v u v2 Derivative Identities n d d n−1 0 c = 0 dx x = 1 dx u = nu u d dx u ud e = u0e d dx d dx u u 0 dx b = ln(b)b u 1 0 d 0 ln u =u u dx logb u = ln b·u u d d 0 0 0 2 sin u = u cos u dx cos u = − u sin u dx tan u = u sec u d dx d dx d d 0 0 2 csc u = − u csc u cot u dx sec u = u sec u tan u dx cot u = − csc u 0 arcsin u =u 2d √ 1 − u dx arccos u = −u0 2d √ 1 − u dx arctan u =u0 d dx 1 + u2 0 arccsc u = −u √ 2 d 0 u u − 1 dx arcsec u =u d dx √ 2 d 0 u u − 1 dx arccot u = −u 1 + u2 Fundamental Theorems of Calculus Zb ( x ) = ⇒ F 0 ( x ) = f a f(x) dx = F(b) − F(a) dx " Z b(x) a(x) d b0(x) − fa(x) f(t) dt = fb(x) · # · a0(x) Integration Rules Linearity Integration by Parts Z Z Z f(x) + g(x) dx = f(x) dx + g(x) dx Z Z u dv = uv − v du Z af(x) dx = a Z f(x) dx Z 0 dx = C Z 1 ZZ Integral Identities Z dx = x + C Z exdx = ex + C xdx = ln |x| + C Z ln x dx = x ln(x) − x + C Z cos x dx = sin x + C n + 1+ C, n 6= −1 n n+1 x dx =x bxdx =bx Z sec x dx = ln |sec x + tan ln(b)[x ln(x) − x] + C Z x| + C 1 sin x dx = − cos x + C Z ZZZZZ sec x tan x dx = sec x + C Z logb x dx =1 tan x dx = − ln | cos x| + C 2 √ 2 x Z a − x dx = arcsin a+ C 1 ln b+ C csc x dx = − ln | csc x + cot x| + C Z 2 csc x cot x dx = − csc x + sec x dx = tan x + C Z 2 C cot x dx = ln |sin x| + C csc x dx = − cot x + C 2 1 x Z a2 + x dx = aarctan a+ C 1 √ 2 x x − 1dx = arcsec x + C Trig Sub a x θ √ 2 2 √ dx = a cos θ dθ a2 − x2 a − x x = a sin θ = a cos θ x = a tan θ dx = a sec2θ dθ θ √ 2 a + x2 = a sec a θ x √ 2 2 a +x √ dθ x2 − a2 = a tan θ √ x x2 − a2 x = a sec θ dx = a sec θ tan θ Source: Stewart, J. (2020). Calculus, 9e. Cengage Learning. θ a