Higher Derivatives
The chain rule given above is obtained by differentiating the identity x = f −1(f(x)) with respect to x. One can continue the same process for higher derivatives. Differentiating the identity with respect to x two times, one obtains
or replacing the first derivative using the formula above,
- .
Similarly for the third derivative:
or using the formula for the second derivative,
These formulas are generalized by the Faà di Bruno's formula.
These formulas can also be written using Lagrange's notation. If f and g are inverses, then
Read more about this topic: Inverse Functions And Differentiation
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—Ralph Waldo Emerson (18031882)