Table of Derivatives

In the following table, a and n are constants, e is the base of the natural logarithms, and u and υ denote functions of x.123 124 1 d d x ( a ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq525.tif"/> 2 d d x ( x ) = 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq526.tif"/> 3 d d x ( a u ) = a d d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq527.tif"/> 4 d d x ( u + υ ) = d u d x + d υ d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq528.tif"/> 5 d d x ( u υ ) = u d υ d x + υ d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq529.tif"/> 6 d d x ( u υ ) = υ d u d x − u d υ d x υ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq530.tif"/> 7 d d x ( u n ) = n u n − 1 d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq531.tif"/> 8 d d x e M = e M d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq532.tif"/> 9 d d x a M = ( log e a ) a M d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq533.tif"/> 10 d d x log e u = ( 1 u ) d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq534.tif"/> 11 d d x log a u = ( log a e ) ( 1 u ) d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq535.tif"/> 12 d d x u v = υ u v − 1 d u d x + u v ( log e u ) d υ d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq536.tif"/> 13 d d x sin u = cos     u d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq537.tif"/> 14 d d x cos u = − sin     u d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq538.tif"/> 15 d d x tan   u = − sec 2   u d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq539.tif"/> 16 d d x ctn  u = − csc 2     u d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq540.tif"/> 17 d d x sec u = sec u tan u d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq541.tif"/> 18 d d x csc u = − csc u   ctn   u d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq542.tif"/> 19 d d x sin − 1 u = 1 1 − u 2 d u d x     ( − 1 2 π ≤ sin − 1 u ≤ 1 2 π ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq543.tif"/> 20 d d x cos − 1 u = − 1 1 − u 2 d u d x     ( 0 ≤ cos − 1 u ≤ π ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq544.tif"/> 21 d d x tan − 1 u = 1 1 + u 2 d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq545.tif"/> 22 d d x ctn − 1 u = − 1 1 + u 2 d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq546.tif"/> 23 d d x sec − 1 u = 1 u u 2 − 1 d u d x ,               ( − π ≤ sec − 1 u < − 1 2 π ; 0 ≤ sec − 1 u < 1 2 π ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq547.tif"/> 24 d d x c sc − 1 u = 1 u u 2 − 1 d u d x ,                 ( − π < csc − 1 u ≤ − 1 2 π ; 0 < csc − 1 u < 1 2 π ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq548.tif"/> 25 d d x sin u = cosh u d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq549.tif"/> 26 d d x cosh u = sinh u d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq550.tif"/> 27 d d x tanh u = sec h 2 u d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq551.tif"/> 28 d d x ctnh   u = − csch 2 u d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq552.tif"/> 29 d d x sech   u = − sech     u     tanh   u d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq553.tif"/> 30 d d x csch   u = csch     u     ctnh   u d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq554.tif"/> 31 d d x sinh − 1 u = 1 u 2 + 1 d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq555.tif"/> 32 d d x cosh − 1 u = 1 u 2 − 1 d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq556.tif"/> 33 d d x tanh − 1 u = 1 1 − u 2 d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq557.tif"/> 34 d d x ctnh − 1   u = − 1 u 2 − 1 d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq558.tif"/> 35 d d x sech − 1 u = − 1 u 1 − u 2 d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq559.tif"/> 36 d d x csch − 1 u = − 1 u u 2 + 1 d u d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203743102/9cca8f32-4688-4a1e-80c2-a6c9068f5153/content/eq560.tif"/>