Derivative of Composite Function/Also presented as

Derivative of Composite Function: Also presented as

The Derivative of Composite Function is also often seen presented using Leibniz's notation for derivatives:

$\dfrac {\d y} {\d x} = \dfrac {\d y} {\d u} \cdot \dfrac {\d u} {\d x}$

or:

$\dfrac \d {\d x} \map u v = \dfrac {\d u} {\d v} \cdot \dfrac {\d v} {\d x}$

where $\dfrac {\d y} {\d x}$ denotes the derivative of $y$ with respect to $x$.


Some sources go so far as to mix their notation and present something like this:

$y' = \dfrac {\d f} {\d g} \map {g'} x$


Also to be mentioned is:

$D_x^1 w = D_u^1 w \, D_x^1 u$

or:

${D_x}^1 w = {D_u}^1 w \, {D_x}^1 u$

where ${D_x}^k u$ denotes the $k$th derivative of $u$ with respect to $x$.


Sources

  • 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Double Function
  • 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 3$. Functions of Several Variables
  • 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: $3.3.5$
  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: General Rules of Differentiation: $13.11$
  • 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $5$. Differential Calculus: Appendix: Differentiation Rules: $5.$ Chain rule
  • 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 10.13$
  • 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: Exercise $21$
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): chain rule (for differentiation)
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): chain rule (for differentiation)
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): chain rule
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): chain rule
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