User:Programming gecko/Derivatives

Nth-order derivatives:

Basic functions

Sum of functions

${\begin{array}{lcl}y&=&u\!\left(x\right)\pm v\!\left(x\right)\\{\frac {d^{n}y}{dx^{n}}}&=&{\frac {d^{n}}{dx^{n}}}u\pm {\frac {d^{n}}{dx^{n}}}v\end{array}}$

Product of functions

${\begin{array}{lcl}y&=&u\!\left(x\right)\cdot v\!\left(x\right)\\{\frac {d^{n}y}{dx^{n}}}&=&\sum _{j=0}^{n}\left({\binom {n}{j}}\cdot {\frac {d^{j}}{dx^{j}}}u\cdot {\frac {d^{n-j}}{dx^{n-j}}}v\right)\end{array}}$

Where ${\frac {d^{0}f}{dx^{0}}}=f$

Quotient of functions

${\begin{array}{lcl}y&=&{\frac {u\left(x\right)}{v\left(x\right)}}\\{\frac {d^{n}y}{dx^{n}}}&=&?\end{array}}$

Composition of functions

${\begin{array}{lcl}y&=&u\!\left(x\right)\circ v\!\left(x\right)=u\!\left(v\!\left(x\right)\right)\\{\frac {d^{n}y}{dx^{n}}}&=&?\end{array}}$

Power function

${\begin{array}{lcl}y&=&x^{m}\\{\frac {d^{n}y}{dx^{n}}}&=&x^{m-n}\cdot \left(\prod _{j=0}^{n-1}\left(n-j\right)\right)\end{array}}$

Trigonometric functions

Sine function

${\begin{array}{lcl}y&=&\sin x\\{\frac {d^{n}y}{dx^{n}}}&=&\sin \left(x+{\frac {n\pi }{2}}\right)\end{array}}$

Cosine function

${\begin{array}{lcl}y&=&\cos x\\{\frac {d^{n}y}{dx^{n}}}&=&\cos \left(x+{\frac {n\pi }{2}}\right)\end{array}}$

Tangent function

${\begin{array}{lcl}y&=&\tan x\\{\frac {d^{n}y}{dx^{n}}}&=&?\end{array}}$

Cotangent function

${\begin{array}{lcl}y&=&\cot x\\{\frac {d^{n}y}{dx^{n}}}&=&?\end{array}}$

Secant function

${\begin{array}{lcl}y&=&\sec x\\{\frac {d^{n}y}{dx^{n}}}&=&?\end{array}}$

Cosecant function

${\begin{array}{lcl}y&=&\csc x\\{\frac {d^{n}y}{dx^{n}}}&=&?\end{array}}$

Exponential functions

Exponential function

${\begin{array}{lcl}y&=&b^{x}\\{\frac {d^{n}y}{dx^{n}}}&=&\left(\ln b\right)^{n}\cdot b^{x}\end{array}}$

Special case

${\begin{array}{lcl}y&=&e^{x}\\{\frac {d^{n}y}{dx^{n}}}&=&e^{x}\end{array}}$