Product rule for higher derivatives
From Calculus
This article is about a differentiation rule, i.e., a rule for differentiating a function expressed in terms of other functions whose derivatives are known.
View other differentiation rules
Statement
Version type | Statement |
---|---|
specific point, named functions | This states that if ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Here, ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
generic point, named functions, point notation | If ![]() ![]() ![]() |
generic point, named functions, point-free notation | If ![]() ![]() ![]() |
Pure Leibniz notation | Suppose ![]() ![]() ![]() ![]() |
One-sided version
There are analogues of each of the statements with one-sided derivatives. Fill this in later
Particular cases
Value of ![]() |
Formula for ![]() |
---|---|
1 | ![]() |
2 | ![]() |
3 | ![]() |
4 | ![]() |
5 | ![]() |
Related rules
Similar rules in single variable calculus
- Product rule for differentiation
- Chain rule for higher derivatives
- Chain rule for differentiation
- Repeated differentiation is linear
Similar rules in multivariable calculus
Similar rules in advanced mathematics
Significance
Qualitative and existential significance
Each of the versions has its own qualitative significance:
Version type | Significance |
---|---|
specific point, named functions | This tells us that if ![]() ![]() ![]() ![]() ![]() |
generic point, named functions, point notation | This tells us that if ![]() ![]() ![]() ![]() |
generic point, named functions, point-free notation | This shows that the way that ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Computational feasibility significance
Each of the versions has its own computational feasibility significance:
Version type | Significance |
---|---|
specific point, named functions | This tells us that knowing the values (in the sense of numerical values) of ![]() ![]() ![]() ![]() |
generic point, named functions | This tells us that knowledge of the generic expressions for ![]() ![]() ![]() |