Table of Derivatives
Following are the derivatives we met in previous chapters:
and this chapter,
1. Powers of x
General formula
`d/dx u^n` `=n u^(n-1) (du)/dx`, where `u` is a function of `x`.
Particular cases and examples
`d/dx c` `=0`
`d/dx x` `=1`
`d/dx x^n` `=n x^(n-1)`
`d/dx x^7` `=7 x^6`
2. Trigonometric Functions
Trigonometry General formulas (a)
`d/dx sin u = (cos u)(du)/dx`
`d/dx cos u = - (sin u) (du)/dx`
`d/dx tan u = (sec^2 u) (du)/dx`
Particular cases and examples
`d/dx sin 3x = 3 cos 3x`
`d/dx sin x^2 =\ 2x\ cos x^2`
`d/dx sin x = cos x`
`d/dx cos x = - sin x`
`d/dx cos^3 x = - 3\ sin^2 x`
`d/dx tan x = sec^2 x`
`d/dx 5tan 7x = 35\ sec^2 7x`
Trigonometry General formulas (b) - reciprocals
`d/dx csc u = (-csc u cot u)(du)/dx`
`d/dx sec u = (sec u tan u)(du)/dx`
`d/dx cot u = (- csc^2 u)(du)/dx`
Particular cases and examples
`d/dx csc x = -csc x cot x`
`d/dx sec x = sec x tan x`
`d/dx cot x = - csc^2 x`
Exponential and Logarithmic Functions
General formulas
`d/dx e^u = (e^u)(du)/dx`
`d/dx b^u = (b^u ln(b))(du)/dx`
`d/dx ln(u) = (1/u)(du)/dx = (u')/u`
Particular cases and examples
`d/dx e^x = e^x`
`d/dx 3^x = 3^x ln(3) = 1.0986 xx 3^x`
`d/dx ln(x) = 1/x`
`d/dx ln(x^4) = 4/x`
`d/dx ln(5x) = 1/x`
Inverse Trigonometric Functions
General formulas
`d/dx arcsin u = (1 / sqrt(1 - u^2))(du)/dx`
`d/dx "arccsc"\ u = (-1 /(|u| sqrt(u^2 - 1)))(du)/dx`
`d/dx arccos u = ( -1 /sqrt(1 - u^2))(du)/dx`
`d/dx "arcsec" u = (1/(|u| sqrt(u^2 - 1)))(du)/dx`
`d/dx arctan u = (1/(1 + u^2))(du)/dx`
`d/dx "arccot"\ u = (-1/(1 + u^2))(du)/dx`
Particular cases
`d/dx arcsin x = 1 / sqrt(1 - x^2)`
`d/dx "arccsc"\ x = -1 /(|x| sqrt(x^2 - 1))`
`d/dx arccos x = -1 /sqrt(1 - x^2)`
`d/dx "arcsec" x = 1/(|x| sqrt(x^2 - 1))`
`d/dx arctan x = 1/(1 + x^2)`
`d/dx "arccot"\ x = -1/(1 + x^2)`
Hyperbolic Functions
The hyperbolic functions are defined as follows:
`sinh x = (e^x-e^(-x))/2`
`cosh x = (e^x+e^(-x))/2`
`tanh x = (sinh x)/(cosh x) = (e^x - e^(-x))/(e^x + e^(-x))`
`"csch"\ x = 1/(sinh x)`
`"sech"\ x = 1/(cosh x)`
`coth x = 1/(tanh x)`
General formulas
`d/dx sinh u = (cosh u )(du)/dx`
`d/dx "csch" u = (- coth u "csch" u)(du)/dx`
`d/dx cosh u = (sinh u)(du)/dx`
`d/dx "sech" u = (- tanh u "sech" u)(du)/dx`
`d/dx "tanh" u = (1 - tanh^2 u)(du)/dx`
`d/dx coth u = (1 - coth^2 u )(du)/dx`
Particular cases
`d/dx sinh x = cosh x`
`d/dx "csch"\ x = - coth x "csch"\ x `
`d/dx cosh x = sinh x `
`d/dx "sech"\ x = - tanh x "sech"\ x `
`d/dx tanh x = 1 - tanh2 x `
`d/dx coth x = 1 - coth2 x `