Chapter 4: Partial Differentiation
Section 4.2: Higher-Order Partial Derivatives
If f=x yx2+y2 and a,b=2,3, obtain all second partial derivatives, both at x,y and at a,b.
Table 4.2.1(a) lists the simplified forms of the second partial derivatives of f, both at x,y and at a,b.
Table 4.2.1(a) Second partials of f at x,y and 2,3
Maple Solution - Interactive
Define f as an expression and use the second-partial operators from the Calculus palette.
Context Panel: Assign Name
Second partial derivatives at x,y
Calculus palette: Second-partials operators
Context Panel: Evaluate and Display Inline
Context Panel: Simplify≻Simplify
∂2∂x2 f = −6⁢x⁢yx2+y22+8⁢x3⁢yx2+y23= simplify 2⁢x⁢y⁢x2−3⁢y2x2+y23
∂2∂ y⁢∂ x f = 1x2+y2−2⁢x2x2+y22−2⁢y2x2+y22+8⁢x2⁢y2x2+y23= simplify −x4−6⁢x2⁢y2+y4x2+y23
∂2∂ x⁢∂ y f = 1x2+y2−2⁢x2x2+y22−2⁢y2x2+y22+8⁢x2⁢y2x2+y23= simplify −x4−6⁢x2⁢y2+y4x2+y23
∂2∂y2 f = −6⁢x⁢yx2+y22+8⁢x⁢y3x2+y23= simplify −2⁢x⁢y⁢3⁢x2−y2x2+y23
Second partial derivatives at a,b
Expression palette: Evaluation template
∂2∂x2 fx=a|f(x)x=2,y=3 = −2762197
∂2∂ y⁢∂ x fx=a|f(x)x=2,y=3 = 1192197
∂2∂y2 fx=a|f(x)x=2,y=3 = −362197
∂2∂ x⁢∂ y fx=a|f(x)x=2,y=3 = 1192197
Alternatively, define the second partial derivatives as functions. Every subscripted name must be an Atomic Identifier.
Calculus palette: Second-partial operators
Context Panel: Assign Function
f__xxx,y=∂2∂x2 f→assign as functionf__xx
f__xyx,y=∂2∂ y⁢∂ x f→assign as functionf__xy
f__yyx,y=∂2∂y2 f→assign as functionf__yy
f__yxx,y=∂2∂ x⁢∂ y f→assign as functionf__yx
Obtain the second partial derivatives at x,y
f__xxx,y = −6⁢y⁢xx2+y22+8⁢x3⁢yx2+y23= simplify 2⁢x⁢y⁢x2−3⁢y2x2+y23
f__xyx,y = 1x2+y2−2⁢x2x2+y22−2⁢y2x2+y22+8⁢x2⁢y2x2+y23= simplify −x4−6⁢x2⁢y2+y4x2+y23
f__yxx,y = 1x2+y2−2⁢x2x2+y22−2⁢y2x2+y22+8⁢x2⁢y2x2+y23= simplify −x4−6⁢x2⁢y2+y4x2+y23
f__yyx,y = −6⁢y⁢xx2+y22+8⁢x⁢y3x2+y23= simplify −2⁢x⁢y⁢3⁢x2−y2x2+y23
Obtain the second partial derivatives at a,b
f__xx2,3 = −2762197
f__xy2,3 = 1192197
f__yx2,3 = 1192197
f__yy2,3 = −362197
Maple Solution - Coded
If f is an expression, the simplified forms for its four second partial derivatives are obtained with the simplify and diff commands. Evaluation at a,b is obtained with the eval command. The names to which the derivatives have been assigned are all Atomic Identifiers.
Assign the expression to the name f.
Second partials at x,y
Second partials at a,b
If f is a function, the simplified forms for its four second partial derivatives are obtained with the simplify command applied to the D-operator. Evaluation at a,b is immediate since the D-operator returns derivatives as functions.
Define the function f.
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