2 1 1 x x 4. u ddx {(x3 + 4x + 1)3/4} = 34 (x3 + 4x + 1)−1/4. •use a table of derivatives, or a table of anti-derivatives, in order to integrate simple func-tions. 1. Quotient Rule v2 vu uv v u dx d ′− ′ = 5. Common Derivatives and Integrals Provided by the Academic Center for Excellence 1 Reviewed June 2008 Common Derivatives and Integrals Derivative Rules: 1. Integrating exponentials 3 4. Table of derivatives Table of integrals 1. Table of derivatives Introduction This leaﬂet provides a table of common functions and their derivatives. A: TABLE OF BASIC DERIVATIVES Let u = u(x) be a differentiable function of the independent variable x, that is u(x) exists. ′(sin x) =cosx 9. (Chain rule) If y = f(u) is differentiable on u = g(x) and u = g(x) is differentiable on point x, then the composite function y = f(g(x)) is differentiable and dx du du dy dx dy = 7. (xα)′=αxα−1 2. If a term in your choice for Yp happens to be a solution of the homogeneous ODE corresponding to (4), multiply this term by x (or by x 2 if this solution corresponds to a double root of the Introduction 2 2. 2. 1. (b) Modification Rule. ′ x x 1 (ln ) = 8. Integrating powers 3 3. Constant Multiple Rule [ ]cu cu dx d = ′, where c is a constant. Sum and Difference Rule [ ]u v u v dx d ± = ±′ 3. (ax)′=ax lna 5. Table 2.1, choose Yp in the same line and determine its undetermined coefficients by substituting Yp and its derivatives into (4). Integrals giving rise to inverse trigonometric functions 5 www.mathcentre.ac.uk 1 c mathcentre 2009. Derivative Table 1. dx dv dx du (u v) dx d ± = ± 2. dx du (cu) c dx d = 3. dx du v dx dv (uv) u dx d = + 4. dx dv wu dx du vw dx dw (uvw) uv dx d = + + 5. v2 dx dv u dx du v v u dx d − = 6. Contents 1. (3x2 + 4)d dx {u} = 12 u.u d dx { 2 − 4x2 + 7x5} = 1 2 2 − 4x2 + 7x5 (−8x + 35x4) d dx {c} = 0 , c is a constant ddx {6} = 0 , since ≅ 3.14 is a constant. ′ x x 2 1 ( ) = 3. Integration Formulas Z dx = x+C (1) Z xn dx = xn+1 n+1 +C (2) Z dx x = ln|x|+C (3) Z ex dx = ex +C (4) Z ax dx = 1 lna ax +C (5) Z lnxdx = xlnx−x+C (6) Z sinxdx = −cosx+C (7) Z cosxdx = sinx+C (8) Z tanxdx = −ln|cosx|+C (9) Z cotxdx = ln|sinx|+C (10) Z secxdx = ln|secx+tanx|+C (11) Z cscxdx = −ln |x+cot +C (12) Z sec2 xdx = tanx+C (13) Z csc2 xdx = −cotx+C (14) Z secxtanxdx = secx+C Integrating trigonmetric functions 4 5. (A) The Power Rule : Examples : d dx {un} = nu n−1. Product Rule [ ]uv uv vu dx d = +′ 4. ( ex)′=ex 6. x a a x ln 1 (log )′= 7.