Editing Lambert W function (section)

== Calculus ==

=== Derivative ===
By [[implicit differentiation]], one can show that all branches of {{mvar|W}} satisfy the [[ordinary differential equation|differential equation]]
: <math>z(1 + W) \frac{dW}{dz} = W \quad \text{for } z \neq -\frac{1}{e}.</math>
({{mvar|W}} is not [[Differentiable function|differentiable]] for {{math|1=''z'' = −{{sfrac|1|''e''}}}}.) As a consequence, that gets the following formula for the derivative of ''W'':
: <math>\frac{dW}{dz} = \frac{W(z)}{z(1 + W(z))} \quad \text{for } z \not\in \left\{0, -\frac{1}{e}\right\}.</math>

Using the identity {{math|1=''e''<sup>''W''(''z'')</sup> = {{sfrac|''z''|''W''(''z'')}}}}, gives the following equivalent formula:
: <math>\frac{dW}{dz} = \frac{1}{z + e^{W(z)}} \quad \text{for } z \neq -\frac{1}{e}.</math>

At the origin we have
: <math>W'_0(0)=1.</math>

The n-th derivative of {{mvar|W}} is of the form:
: <math>\frac{d^{n}W}{dz^{n}} = \frac{P_{n}(W(z))}{(z + e^{W(z)})^{n}(W(z) + 1)^{n - 1}} \quad \text{for } n > 0,\, z \ne -\frac{1}{e}.</math>

Where {{math|''P<sub>n</sub>''}} is a polynomial function with coefficients defined in {{OEIS link|A042977}}. If and only if {{mvar|z}} is a root of {{math|''P<sub>n</sub>''}} then {{math|''ze<sup>z</sup>''}} is a root of the n-th derivative of {{mvar|W}}.

Taking the derivative of the n-th derivative of {{mvar|W}} yields:
: <math>\frac{d^{n + 1}W}{dz^{n + 1}} = \frac{(W(z) + 1)P_{n}'(W(z)) + (1 - 3n - nW(z))P_{n}(W(z))}{(n + e^{W(z)})^{n + 1}(W(z) + 1)^{n}} \quad \text{for } n > 0,\, z \ne -\frac{1}{e}.</math>
Inductively proving the n-th derivative equation.

=== Integral ===
The function {{math|''W''(''x'')}}, and many other expressions involving {{math|''W''(''x'')}}, can be [[integral|integrated]] using the [[substitution rule|substitution]] {{math|1=''w'' = ''W''(''x'')}}, i.e. {{math|1=''x'' = ''we''<sup>''w''</sup>}}:
: <math> \begin{align}
\int W(x)\,dx &= x W(x) - x + e^{W(x)} + C\\
& = x \left( W(x) - 1 + \frac{1}{W(x)} \right) + C.
\end{align}</math>
(The last equation is more common in the literature but is undefined at {{math|1=''x'' = 0}}). One consequence of this (using the fact that {{math|1=''W''<sub>0</sub>(''e'') = 1}}) is the identity
: <math>\int_{0}^{e} W_0(x)\,dx = e - 1.</math>