Chapter 4: Partial Differentiation
Section 4.1: First-Order Partial Derivatives
Example 4.1.1
If f=x siny+y sinx and a,b=π/3,π/6, obtain fx and fy both at x,y and at a,b.
Solution
Mathematical Solution
The requisite calculations are summarized in Table 4.1.1(a).
∂∂xx siny+y sinx=siny+ycosx
∂∂yx siny+y sinx=xcosy+sinx
∂f∂xx=a|f(x)x,y=π/3,π/6 = 12+112π
∂f∂yx=a|f(x)x,y=π/3,π/6= 32π3+1
Table 4.1.1(a) First partial derivatives of f
Maple Solution - Interactive
Calculating partial derivatives and evaluating them at a point can be done with just the Context Panel system.
Context Panel
Control-drag the expression for f and press the Enter key.
Context Panel: Differentiate≻With Respect To≻x (or y)
Context Panel: Evaluate at a Point (see Figure 4.1.1(a)).
Figure 4.1.1(a) Evaluate at a,b
fx
fy
x siny+y sinx
xsiny+ysinx
→differentiate w.r.t. x
siny+ycosx
→evaluate at point
12+112π
→differentiate w.r.t. y
xcosy+sinx
16π3+123
Defining f as an expression allows its partial derivatives to be calculated and evaluated at a point via some of the palette templates, allowing for a more natural notation to be displayed.
Define f as an expression
Control-drag the expression for f.
Context Panel: Assign to a Name≻f
x siny+y sinx→assign to a namef
Obtain fxx,y and fyx,y
Calculus palette: First-partial operator
Context Panel: Evaluate and Display Inline
∂∂ x f = siny+ycosx
∂∂ y f = xcosy+sinx
Obtain fxa,b and fya,b
Expression palette: Evaluation template Calculus palette: First-partial operator
∂∂ x fx=a|f(x)x=π/3,y=π/6 = 12+112π
∂∂ y fx=a|f(x)x=π/3,y=π/6 = 16π3+123
A very high degree of notational faithfulness can be obtained by defining subscripts as operators.
In the present context, the expression for f is already assigned to the name f. Were this not so, the expression would have to be assigned to a name, preferably, f.
Define the functions fx and fy
Write the symbols fx and fy as Atomic Identifiers.
Context Panel: Assign Function
f__xx,y=∂∂ x f→assign as functionf__x
f__yx,y=∂∂ y f→assign as functionf__y
f__xx,y = siny+ycosx
f__yx,y = xcosy+sinx
f__xπ/3,π/6 = 12+112π
f__yπ/3,π/6 = 16π3+123
Maple Solution - Coded
Assign to f and press the Enter key.
f≔x siny+y sinx:
Apply the diff command and press the Enter key.
fx≔difff,x
fy≔difff,y
Apply the eval command and press the Enter key.
evalfx,x=π/3,y=π/6
evalfy,x=π/3,y=π/6
Alternatively, define f as a function.
Define f as a function
Use the arrow notation to define f.
f≔x,y→x siny+y sinx:
Use the D-operator to obtain fxx,y.
D1fx,y = siny+ycosx
Use the D-operator to obtain fyx,y.
D2fx,y = xcosy+sinx
Use the D-operator to obtain fxa,b.
D1fπ/3,π/6 = 12+112π
Use the D-operator to obtain fya,b.
D2fπ/3,π/6 = 16π3+123
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