Difference between revisions of "The Chain Rule for Functions of more than One Variable"
Jump to navigation
Jump to search
(Created page with "*[https://www.youtube.com/watch?v=XipB_uEexF0 Chain Rule With Partial Derivatives - Multivariable Calculus] Video by The Organic Chemistry Tutor 2019 *") |
|||
Line 1: | Line 1: | ||
+ | ===Chain Rule for One Independent Variable=== | ||
+ | Suppose that <math>x=g(t)</math> and <math>y=h(t)</math> are differentiable functions of <math>t</math> and <math>z=f(x,y)</math> is a differentiable function of <math>x</math> and <math>y</math>. Then <math>z=f(x(t),y(t))</math> is a differentiable function of <math>t</math> and | ||
+ | |||
+ | <math>\frac{dz}{dt} = \frac{\partial z}{\partial x}\cdot \frac{dx}{dt} + \frac{\partial z}{\partial y}\cdot \frac{dy}{dt} </math> | ||
+ | |||
+ | where the ordinary derivatives are evaluated at t and the partial derivatives are evaluated at (x,y). | ||
+ | |||
+ | === Rules of taking Jacobians === | ||
+ | If '''''f''''' : '''R'''<sup>''m''</sup> → '''R'''<sup>''n''</sup>, and ''h''(''x'') : '''R'''<sup>''m''</sup> → '''R''' are differentiable at ''''p'''': | ||
+ | * <math>J_\mathbf{p} (\mathbf{f}+\mathbf{g}) = J_\mathbf{p} \mathbf{f} + J_\mathbf{p} \mathbf{g}</math> | ||
+ | * <math>J_\mathbf{p} (h\mathbf{f}) = hJ_\mathbf{p} \mathbf{f} + \mathbf{f}(\mathbf{p}) J_\mathbf{p} h</math> | ||
+ | * <math>J_\mathbf{p} (\mathbf{f}\cdot \mathbf{g}) = \mathbf{g}^T J_\mathbf{p} \mathbf{f} + \mathbf{f}^T J_\mathbf{p}\mathbf{g}</math> | ||
+ | Important: make sure the order is right - matrix multiplication is not commutative! | ||
+ | |||
+ | ==== Chain rule with Jacobian ==== | ||
+ | The chain rule for functions of several variables is as follows. For '''''f''''' : '''R'''<sup>''m''</sup> → '''R'''<sup>''n''</sup> and '''''g''''' : '''R'''<sup>''n''</sup> → '''R'''<sup>''p''</sup>, and '''''g''''' o '''''f''''' differentiable at '''''p''''', then the Jacobian is given by | ||
+ | : <math>\left( J_{\mathbf{f}(\mathbf{p})} \mathbf{g}\right) \left( J_\mathbf{p} \mathbf{f}\right)</math> | ||
+ | Again, we have matrix multiplication, so one must preserve this exact order. | ||
+ | Compositions in one order may be defined, but not necessarily in the other way. | ||
+ | |||
+ | ==Resources== | ||
+ | * [https://en.wikibooks.org/wiki/Calculus/Multivariable_Calculus/Chain_Rule Chain Rule], WikiBooks: Multivariable Calculus | ||
+ | * [https://math.hmc.edu/calculus/hmc-mathematics-calculus-online-tutorials/multivariable-calculus/multi-variable-chain-rule/ Multivariable Chain Rule], Harvey Mudd College | ||
*[https://www.youtube.com/watch?v=XipB_uEexF0 Chain Rule With Partial Derivatives - Multivariable Calculus] Video by The Organic Chemistry Tutor 2019 | *[https://www.youtube.com/watch?v=XipB_uEexF0 Chain Rule With Partial Derivatives - Multivariable Calculus] Video by The Organic Chemistry Tutor 2019 | ||
− | |||
− |
Revision as of 12:13, 6 October 2021
Contents
Chain Rule for One Independent Variable
Suppose that and are differentiable functions of and is a differentiable function of and . Then is a differentiable function of and
where the ordinary derivatives are evaluated at t and the partial derivatives are evaluated at (x,y).
Rules of taking Jacobians
If f : Rm → Rn, and h(x) : Rm → R are differentiable at 'p':
Important: make sure the order is right - matrix multiplication is not commutative!
Chain rule with Jacobian
The chain rule for functions of several variables is as follows. For f : Rm → Rn and g : Rn → Rp, and g o f differentiable at p, then the Jacobian is given by
Again, we have matrix multiplication, so one must preserve this exact order. Compositions in one order may be defined, but not necessarily in the other way.
Resources
- Chain Rule, WikiBooks: Multivariable Calculus
- Multivariable Chain Rule, Harvey Mudd College
- Chain Rule With Partial Derivatives - Multivariable Calculus Video by The Organic Chemistry Tutor 2019