Example 4-1-1 - Maple Help

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Chapter 4: Partial Differentiation



Section 4.1: First-Order Partial Derivatives



Example 4.1.1



 If  and $\left(a,b\right)=\left(\mathrm{π}/3,\mathrm{π}/6\right)$, obtain ${f}_{x}$ and ${f}_{y}$ both at $\left(x,y\right)$ and at $\left(a,b\right)$.



Solution

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 $\left(a,b\right)$

${f}_{x}$

${f}_{y}$

${x}{}{\mathrm{sin}}{}\left({y}\right){+}{y}{}{\mathrm{sin}}{}\left({x}\right)$

$\stackrel{\text{differentiate w.r.t. x}}{\to }$

${\mathrm{sin}}{}\left({y}\right){+}{y}{}{\mathrm{cos}}{}\left({x}\right)$

$\stackrel{\text{evaluate at point}}{\to }$

$\frac{{1}}{{2}}{+}\frac{{1}}{{12}}{}{\mathrm{π}}$

${x}{}{\mathrm{sin}}{}\left({y}\right){+}{y}{}{\mathrm{sin}}{}\left({x}\right)$

$\stackrel{\text{differentiate w.r.t. y}}{\to }$

${x}{}{\mathrm{cos}}{}\left({y}\right){+}{\mathrm{sin}}{}\left({x}\right)$

$\stackrel{\text{evaluate at point}}{\to }$

$\frac{{1}}{{6}}{}{\mathrm{π}}{}\sqrt{{3}}{+}\frac{{1}}{{2}}{}\sqrt{{3}}$



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≻$f$

$\stackrel{\text{assign to a name}}{\to }$${f}$

Obtain ${f}_{x}\left(x,y\right)$ and ${f}_{y}\left(x,y\right)$

 • Calculus palette: First-partial operator
 • Context Panel: Evaluate and Display Inline

= ${\mathrm{sin}}{}\left({y}\right){+}{y}{}{\mathrm{cos}}{}\left({x}\right)$

 • Calculus palette: First-partial operator
 • Context Panel: Evaluate and Display Inline

= ${x}{}{\mathrm{cos}}{}\left({y}\right){+}{\mathrm{sin}}{}\left({x}\right)$

Obtain ${f}_{x}\left(a,b\right)$ and ${f}_{y}\left(a,b\right)$

 • Expression palette: Evaluation template Calculus palette: First-partial operator
 • Context Panel: Evaluate and Display Inline

= $\frac{{1}}{{2}}{+}\frac{{1}}{{12}}{}{\mathrm{π}}$

 • Expression palette: Evaluation template Calculus palette: First-partial operator
 • Context Panel: Evaluate and Display Inline

= $\frac{{1}}{{6}}{}{\mathrm{π}}{}\sqrt{{3}}{+}\frac{{1}}{{2}}{}\sqrt{{3}}$



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  ${f}_{x}$ and ${f}_{y}$

 • Write the symbols ${f}_{x}$ and ${f}_{y}$ as Atomic Identifiers.
 • Calculus palette: First-partial operator
 • Context Panel: Assign Function

${f}_{x}$

${f}_{y}$

$\stackrel{\text{assign as function}}{\to }$$\mathrm{f__x}$

$\stackrel{\text{assign as function}}{\to }$$\mathrm{f__y}$

Obtain ${f}_{x}\left(x,y\right)$ and ${f}_{y}\left(x,y\right)$

$\mathrm{f__x}\left(x,y\right)$ = ${\mathrm{sin}}{}\left({y}\right){+}{y}{}{\mathrm{cos}}{}\left({x}\right)$

$\mathrm{f__y}\left(x,y\right)$ = ${x}{}{\mathrm{cos}}{}\left({y}\right){+}{\mathrm{sin}}{}\left({x}\right)$

Obtain ${f}_{x}\left(a,b\right)$ and ${f}_{y}\left(a,b\right)$

$\mathrm{f__x}\left(\mathrm{π}/3,\mathrm{π}/6\right)$ = $\frac{{1}}{{2}}{+}\frac{{1}}{{12}}{}{\mathrm{π}}$

$\mathrm{f__y}\left(\mathrm{π}/3,\mathrm{π}/6\right)$ = $\frac{{1}}{{6}}{}{\mathrm{π}}{}\sqrt{{3}}{+}\frac{{1}}{{2}}{}\sqrt{{3}}$







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