Editing Inverse function rule (section)

==Higher derivatives==

The [[chain rule]] given above is obtained by differentiating the identity <math>f^{-1}(f(x))=x</math> with respect to {{Mvar|x}}. One can continue the same process for higher derivatives. Differentiating the identity twice with respect to ''{{Mvar|x}}'', one obtains

:<math> \frac{d^2y}{dx^2}\,\cdot\,\frac{dx}{dy} + \frac{d}{dx} \left(\frac{dx}{dy}\right)\,\cdot\,\left(\frac{dy}{dx}\right)  =  0, </math>

that is simplified further by the chain rule as

:<math> \frac{d^2y}{dx^2}\,\cdot\,\frac{dx}{dy} + \frac{d^2x}{dy^2}\,\cdot\,\left(\frac{dy}{dx}\right)^2  =  0.</math>

Replacing the first derivative, using the identity obtained earlier, we get

:<math> \frac{d^2y}{dx^2} = - \frac{d^2x}{dy^2}\,\cdot\,\left(\frac{dy}{dx}\right)^3. </math>

Similarly for the third derivative:

:<math> \frac{d^3y}{dx^3} = - \frac{d^3x}{dy^3}\,\cdot\,\left(\frac{dy}{dx}\right)^4 -
3 \frac{d^2x}{dy^2}\,\cdot\,\frac{d^2y}{dx^2}\,\cdot\,\left(\frac{dy}{dx}\right)^2</math>

or using the formula for the second derivative,

:<math> \frac{d^3y}{dx^3} = - \frac{d^3x}{dy^3}\,\cdot\,\left(\frac{dy}{dx}\right)^4 +
3 \left(\frac{d^2x}{dy^2}\right)^2\,\cdot\,\left(\frac{dy}{dx}\right)^5</math>

These formulas are generalized by the [[Faà di Bruno's formula]].

These formulas can also be written using Lagrange's notation. If ''{{Mvar|f}}'' and ''{{Mvar|g}}'' are inverses, then

:<math> g''(x) = \frac{-f''(g(x))}{[f'(g(x))]^3}</math>