′ x x 1 (ln ) = 8. Constant Multiple Rule [ ]cu cu dx d = ′, where c is a constant. •use a table of derivatives, or a table of anti-derivatives, in order to integrate simple func-tions. Product Rule [ ]uv uv vu dx d = +′ 4. 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. 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

Introduction 2 2. ( ex)′=ex 6. x a a x ln 1 (log )′= 7. 1.

(b) Modification Rule. 1. Integrating exponentials 3 4. Table of derivatives Table of integrals 1. Quotient Rule v2 vu uv v u dx d ′− ′ = 5. (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. u ddx {(x3 + 4x + 1)3/4} = 34 (x3 + 4x + 1)−1/4. Table of derivatives Introduction This leaflet provides a table of common functions and their derivatives. (xα)′=αxα−1 2. Common Derivatives and Integrals Provided by the Academic Center for Excellence 1 Reviewed June 2008 Common Derivatives and Integrals Derivative Rules: 1.

A: TABLE OF BASIC DERIVATIVES Let u = u(x) be a differentiable function of the independent variable x, that is u(x) exists. Table 2.1, choose Yp in the same line and determine its undetermined coefficients by substituting Yp and its derivatives into (4). ′(sin x) =cosx 9. Integrating powers 3 3. (A) The Power Rule : Examples : d dx {un} = nu n−1. ′ x x 2 1 ( ) = 3. 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 (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.

Integrals giving rise to inverse trigonometric functions 5 www.mathcentre.ac.uk 1 c mathcentre 2009. Sum and Difference Rule [ ]u v u v dx d ± = ±′ 3.

2. Integrating trigonmetric functions 4 5. Contents 1. 2 1 1 x x 4. (ax)′=ax lna 5.