Product Rule for Derivatives
Theorem
Let $\map f x, \map j x, \map k x$ be real functions defined on the open interval $I$.
Let $\xi \in I$ be a point in $I$ at which both $j$ and $k$ are differentiable.
Let $\map f x = \map j x \map k x$.
Then:
- $\map {f'} \xi = \map j \xi \map {k'} \xi + \map {j'} \xi \map k \xi$
It follows from the definition of derivative that if $j$ and $k$ are both differentiable on the interval $I$, then:
- $\forall x \in I: \map {f'} x = \map j x \map {k'} x + \map {j'} x \map k x$
Using Leibniz's notation for derivatives, this can be written as:
- $\map {\dfrac \d {\d x} } {y \, z} = y \dfrac {\d z} {\d x} + \dfrac {\d y} {\d x} z$
where $y$ and $z$ represent functions of $x$.
General Result
Let $\map {f_1} x, \map {f_2} x, \ldots, \map {f_n} x$ be real functions differentiable on the open interval $I$.
then:
- $\forall x \in I: \ds \map {D_x} {\prod_{i \mathop = 1}^n \map {f_i} x} = \sum_{i \mathop = 1}^n \paren {\map {D_x} {\map {f_i} x} \prod_{j \mathop \ne i} \map {f_j} x}$
Proof
First we note that from Differentiable Function is Continuous‎, $j$ is continuous at $\xi$.
Hence:
- $(1): \quad \map j {\xi + h} \to \map j \xi$ as $h \to 0$
So:
| \(\ds \map {f'} \xi\) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h\) | Definition of Derivative | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\map j {\xi + h} \map k {\xi + h} - \map j \xi \map k \xi} h\) | by hypothesis | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\map j {\xi + h} \map k {\xi + h} - \map j {\xi + h} \map k \xi + \map j {\xi + h} \map k \xi - \map j \xi \map k \xi} h\) | adding $\pm \map j {\xi + h} \map k \xi$ to numerator | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \paren {\map j {\xi + h} \frac {\map k {\xi + h} - \map k \xi} h + \frac {\map j {\xi + h} - \map j \xi} h \map k \xi}\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \paren {\map j {\xi + h} \frac {\map k {\xi + h} - \map k \xi} h} + \lim_{h \mathop \to 0} \paren {\frac {\map j {\xi + h} - \map j \xi} h \map k \xi}\) | Sum Rule for Limits of Real Functions | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \paren {\map j {\xi + h} } \lim_{h \mathop \to 0} \paren {\frac {\map k {\xi + h} - \map k \xi} h} + \lim_{h \mathop \to 0} \paren {\frac {\map j {\xi + h} - \map j \xi} h} \lim_{h \mathop \to 0} \paren {\map k \xi}\) | Product Rule for Limits of Real Functions | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \paren {\map j {\xi + h} } \map {k'} \xi + \map {j'} \xi \lim_{h \mathop \to 0} \paren {\map k \xi}\) | Definition of Derivative | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map j \xi \map {k'} \xi + \map {j'} \xi \map k \xi\) | from $(1)$ |
$\blacksquare$
Examples
Example: $2 a x e^{a x^2}$
- $\map {\dfrac \d {\d x} } {2 a x e^{a x^2} } = 2 a e^{a x^2} \paren {1 + 2 a x^2}$
Example: $x \sin x$
- $\map {\dfrac \d {\d x} } {x \sin x} = x \cos x + \sin x$
Example: $x \cot x$
- $\map {\dfrac \d {\d x} } {x \cot x} = \cot x - x \cosec^2 x$
Example: $x^2 \arctan x$
- $\map {\dfrac \d {\d x} } {x^2 \arctan x} = 2 x \arctan x + \dfrac {x^2} {1 + x^2}$
Example: $x e^x \sin x$
- $\map {\dfrac \d {\d x} } {x e^x \sin x} = e^x \paren {\paren {1 + x} \sin x + x \cos x}$
Also see
- Derivative of Product of Real Function and Vector-Valued Function
- Derivative of Vector Cross Product of Vector-Valued Functions
- Derivative of Dot Product of Vector-Valued Functions
- Derivative of Product of Operator-Valued Functions
- Leibniz's Rule in One Variable, of which this is the special case of the first derivative
Historical Note
The was first obtained by Gottfried Wilhelm von Leibniz in $1677$.
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.3$ Rules for Differentiation and Integration: Derivatives: $3.3.3$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: General Rules of Differentiation: $13.7$
- 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $5$. Differential Calculus: Appendix: Differentiation Rules: $3.$ Product rule
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Leibniz theorem
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): product rule (for differentiation)
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $1$: Derivatives
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Leibniz theorem
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): product rule (for differentiation)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $1$: Derivatives