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


 * Quotient of functions:

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


 * Power function:

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


 * Cosine function

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


 * Tangent function

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


 * Cotangent function

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


 * Secant function

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


 * Cosecant function

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

Exponential functions

 * Exponential function

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


 * Special case

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