Chapter 4: Partial Differentiation
Section 4.3: Chain Rule
The composition of fx,y=3−x2−y2 with xt=t,yt=t2 forms the function Ft=fxt,yt. Obtain F′t by an appropriate form of the chain rule, and again by writing the rule for F explicitly. Give a graphical interpretation of Ft.
An application of the chain rule gives
=fx x′t+fy y′t
=−2 x⋅1+−2 y⋅2 t
=−2 t⋅1+−2 t2⋅2 t
=−2 t−4 t3
Writing Ft=3−t2−t22=3−t2−t4 explicitly gives F′t=−2 t−4 t3, in agreement with the chain-rule result.
Maple Solution - Interactive
Formal statement of the relevant chain rule
Context Panel: Differentiate≻With Respect To≻t
fxt,yt→differentiate w.r.t. tD1⁡f⁡x⁡t,y⁡t⁢ⅆⅆt⁢x⁡t+D2⁡f⁡x⁡t,y⁡t⁢ⅆⅆt⁢y⁡t
It is possible to obtain notational simplifications interactively, via the Typesetting Rules Assistant in the View menu. However, this is a tedious multistep process, so will not be pursued here.
Implement the chain rule
Context Panel: Assign Function
fx,y=3−x2−y2→assign as functionf
Calculus palette: Partial and ordinary differential operators
Context Panel: Evaluate at a Point≻x=t,y=t2
∂∂ x fx,y ⅆⅆ t t+∂∂ y fx,y ⅆⅆ t t2 = −4⁢t⁢y−2⁢x→evaluate at point−4⁢t3−2⁢t
Obtain F′t from an explicit representation of Ft
Write the explicit form of Ft.
Press the Enter key.
Differentiate≻With Respect To≻t
→differentiate w.r.t. t
Maple Solution - Coded
Simplified Maple notation is available if the commands to the right are first executed.
Although the chain rule for this problem could be written as F′t=fx x′+fy y′, Maple uses the D-operator notation to express the partial derivatives fx and fy, and cannot suppress the arguments of f once suppression of arguments has been applied to x and y.
Restore the variables x and y.
Define the function f.
Obtain the derivative by applying the chain rule.
D1ft,t2 difft,t+D2ft,t2 difft2,t
Define the expression for Ft.
<< Chapter Overview Section 4.3
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2023. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
What kind of issue would you like to report? (Optional)