Leibniz rule
E582439
The Leibniz rule is a fundamental property of derivatives stating that the derivative of a product equals the sum of each factor’s derivative times the other factor.
Statements (39)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
property of derivatives ⓘ rule in calculus ⓘ |
| appliesTo |
C^n functions
ⓘ
derivative of a product ⓘ differentiable functions ⓘ smooth functions ⓘ |
| category | differentiation rule ⓘ |
| expresses | product rule for derivatives ⓘ |
| field |
calculus
ⓘ
mathematical analysis ⓘ |
| generalizes | product rule for first derivatives ⓘ |
| hasConsequence |
derivative of a constant times a function equals the constant times the derivative of the function
ⓘ
derivative of x^n can be computed by repeated application of the product rule ⓘ |
| hasFormula | (f g)' = f' g + f g' ⓘ |
| hasGeneralForm | d^n(fg)/dx^n = Σ_{k=0}^n (n choose k) f^{(k)} g^{(n-k)} ⓘ |
| hasNotation | (fg)^{(n)} = Σ_{k=0}^n (n choose k) f^{(k)} g^{(n-k)} ⓘ |
| hasVariant | Leibniz integral rule NERFINISHED ⓘ |
| holdsFor |
complex-valued differentiable functions
ⓘ
differentiable vector-valued functions ⓘ differential operators ⓘ real-valued differentiable functions ⓘ |
| involves |
binomial coefficients
ⓘ
higher-order derivatives ⓘ |
| isEquivalentTo | product rule in elementary calculus ⓘ |
| logicalType | universal statement about differentiable functions ⓘ |
| namedAfter | Gottfried Wilhelm Leibniz NERFINISHED ⓘ |
| relatedTo |
chain rule
ⓘ
linearity of differentiation ⓘ |
| requires | existence of derivatives of the factors ⓘ |
| states | the derivative of a product equals the sum of each factor’s derivative times the other factor ⓘ |
| usedIn |
Taylor series expansions
ⓘ
differential equations ⓘ distribution theory ⓘ functional analysis ⓘ multivariable calculus ⓘ operator calculus ⓘ |
| usedToProve |
Leibniz formula for higher derivatives of products
NERFINISHED
ⓘ
properties of polynomial derivatives ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.