{"id":1083,"date":"2025-10-15T14:04:20","date_gmt":"2025-10-15T14:04:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&p=1083"},"modified":"2025-10-27T12:57:37","modified_gmt":"2025-10-27T12:57:37","slug":"introduction-to-partial-derivatives-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/introduction-to-partial-derivatives-learn-it-3\/","title":{"raw":"Introduction to Partial Derivatives: Learn It 3","rendered":"Introduction to Partial Derivatives: Learn It 3"},"content":{"raw":"

Higher-Order Partial Derivatives<\/h2>\r\n

Consider the function<\/p>\r\n

[latex]f(x,y)=2x^{3}-4xy^{2}+5y^{3}-6xy+5x-4y+12[\/latex].<\/p>\r\n

Its partial derivatives are<\/p>\r\n

[latex]\\dfrac{\\partial f}{\\partial x}=6x^{2}-4y^{2}-6y+5[\/latex] and [latex]\\dfrac{\\partial f}{\\partial y}=-8xy+15y^{2}-6x-4[\/latex].<\/p>\r\n

Each of these partial derivatives is itself a function of two variables, so we can calculate partial derivatives of these functions. Just as with single-variable derivatives, we call these second-order derivatives, third-order derivatives, and so on. In general, they are higher-order partial derivatives<\/strong>.<\/p>\r\n

There are four second-order partial derivatives for any function (provided they all exist):<\/p>\r\n

[latex]\\large{\\frac{\\partial^{2}f}{\\partial x^{2}}=\\frac{\\partial}{\\partial x}\\left[\\frac{\\partial f}{\\partial x}\\right],\\ \\frac{\\partial^{2}f}{\\partial x\\partial y}=\\frac{\\partial}{\\partial x}\\left[\\frac{\\partial f}{\\partial y}\\right],\\ \\frac{\\partial^{2}f}{\\partial y\\partial x}=\\frac{\\partial}{\\partial y}\\left[\\frac{\\partial f}{\\partial x}\\right],\\ \\frac{\\partial^{2}f}{\\partial y^{2}}=\\frac{\\partial}{\\partial y}\\left[\\frac{\\partial f}{\\partial y}\\right]}.[\/latex]<\/p>\r\n

An alternative notation for these is [latex]f_{xx}[\/latex], [latex]f_{yx}[\/latex], [latex]f_{xy}[\/latex], and [latex]f_{yy}[\/latex], respectively. Higher-order partial derivatives calculated with respect to different variables, such as [latex]f_{xy}[\/latex] and [latex]f_{yx}[\/latex], are called mixed partial derivatives<\/strong>.<\/p>\r\n\r\n

\r\n

In [latex]\\frac{\\partial^{2}f}{\\partial y\\partial x}[\/latex], read the variables from right to left: differentiate first with respect to [latex]x[\/latex], then with respect to [latex]y[\/latex].<\/p>\r\n

In subscript notation [latex]f_{xy}[\/latex], read left to right: differentiate first with respect to [latex]x[\/latex], then with respect to [latex]y[\/latex].<\/p>\r\n\r\n<\/section>

Calculate all four second partial derivates for the function\r\n
[latex]\\large{f\\,(x,\\ y)=xe^{-3y}+\\sin{(2x-5y)}}[\/latex].<\/div>\r\n[reveal-answer q=\"fs-id1367983930124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1367983930124\"]\r\n

To calculate [latex]\\partial^{2}f\/\\partial x^{2}[\/latex] and [latex]\\partial^{2}f\/\\partial y\\partial x[\/latex], we first calculate [latex]\\partial f\/\\partial x[\/latex]:<\/p>\r\n\r\n

[latex]\\large{\\frac{\\partial f}{\\partial x}=e^{-3y}+2\\cos{(2x-5y)}}.[\/latex]<\/div>\r\n \r\n\r\nTo calculate [latex]\\partial^{2}f\/\\partial x^{2}[\/latex], differentiate [latex]\\partial f\/\\partial x[\/latex] with respect to [latex]x[\/latex]:\r\n
[latex]\\begin{array}{ccc}\\hfill {\\frac{{\\partial}^{2} f}{\\partial {x^{2}}}} & =\\hfill & {\\frac{\\partial}{\\partial x}\\left[\\frac{\\partial f}{\\partial x}\\right]} \\hfill \\\\ \\hfill & =\\hfill & {\\frac{\\partial}{\\partial x}\\left[e^{-3y}+2\\cos{(2x-5y)}\\right]} \\hfill \\\\ \\hfill & =\\hfill & {-4\\sin{(2x-5y)}.} \\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n \r\n\r\nTo calculate [latex]\\partial^{2}f\/\\partial y\\partial x[\/latex], differentiate [latex]\\partial f\/\\partial x[\/latex] with respect to [latex]y[\/latex]:\r\n
[latex]\\begin{array}{ccc}\\hfill {\\frac{\\partial^{2}f}{\\partial y\\partial x}} & =\\hfill & {\\frac{\\partial}{\\partial y}\\left[\\frac{\\partial f}{\\partial x}\\right]} \\hfill \\\\ \\hfill & =\\hfill & {\\frac{\\partial}{\\partial y}\\left[e^{-3y}+2\\cos{(2x-5y)}\\right]} \\hfill \\\\ \\hfill & =\\hfill & {-3e^{-3y}+10\\sin{(2x-5y)}.} \\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n \r\n\r\nTo calculate [latex]\\partial^{2}f\/\\partial x\\partial y[\/latex] and [latex]\\partial^{2}f\/\\partial y^{2}[\/latex], first calculate [latex]\\partial f\/\\partial y[\/latex]:\r\n
[latex]\\large{\\frac{\\partial f}{\\partial y}=-3xe^{-3y}-5\\cos{(2x-5y)}.}[\/latex]<\/div>\r\n \r\n\r\nTo calculate [latex]\\partial^{2}f\/\\partial x\\partial y[\/latex], differentiate [latex]\\partial f\/\\partial y[\/latex] with respect to [latex]x[\/latex]:\r\n
[latex]\\begin{array}{ccc}\\hfill {\\frac{\\partial^{2}f}{\\partial x\\partial y}} & =\\hfill & {\\frac{\\partial}{\\partial x}\\left[\\frac{\\partial f}{\\partial y}\\right]} \\hfill \\\\ \\hfill & =\\hfill & {\\frac{\\partial}{\\partial x}\\left[-3xe^{-3y}-5\\cos{(2x-5y)}\\right]} \\hfill \\\\ \\hfill & =\\hfill & {-3e^{-3y}+10\\sin{(2x-5y)}.} \\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n \r\n\r\nTo calculate [latex]\\partial^{2}f\/\\partial^{2} y[\/latex], differentiate [latex]\\partial f\/\\partial y[\/latex] with respect to [latex]y[\/latex]:\r\n
[latex]\\begin{array}{ccc}\\hfill {\\frac{\\partial^{2}f}{\\partial y}} & =\\hfill & {\\frac{\\partial}{\\partial y}\\left[\\frac{\\partial f}{\\partial y}\\right]} \\hfill \\\\ \\hfill & =\\hfill & {\\frac{\\partial}{\\partial y}\\left[-3xe^{-3y}-5\\cos{(2x-5y)}\\right]} \\hfill \\\\ \\hfill & =\\hfill & {9xe^{-3y}-25\\sin{(2x-5y)}.} \\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>Notice something interesting in the previous example: [latex]\\dfrac{\\partial^{2}f}{\\partial x\\partial y}=\\dfrac{\\partial^{2}f}{\\partial y\\partial x}[\/latex]. Both mixed partial derivatives equal [latex]-3e^{-3y}+10\\sin(2x-5y)[\/latex]. Is this always true? Under certain conditions, yes. In fact, this is guaranteed by the following theorem.\r\n\r\n
\r\n

equality of mixed partial derivatives (Clairaut's Theorem)<\/h3>\r\nSuppose [latex]f\\,(x,\\ y)[\/latex] is defined on an open disk [latex]D[\/latex] that contains the point [latex](a,\\ b)[\/latex]. If the functions [latex]f_{xy}[\/latex] and [latex]f_{yx}[\/latex] are continuous on [latex]D[\/latex], then [latex]f_{xy}=f_{yx}[\/latex].\r\n\r\n<\/section>\r\n

Clairaut's theorem guarantees that as long as mixed second-order derivatives are continuous, the order in which we differentiate doesn't matter. We can differentiate with respect to [latex]x[\/latex] first then [latex]y[\/latex], or [latex]y[\/latex] first then [latex]x[\/latex]\u2014we'll get the same result. This theorem can be extended to higher-order derivatives as well.<\/p>\r\n

Beyond the mixed partial derivatives, we also have [latex]f_{xx}[\/latex] (the partial derivative of [latex]f_x[\/latex] with respect to [latex]x[\/latex]) and [latex]f_{yy}[\/latex] (the partial derivative of [latex]f_y[\/latex] with respect to [latex]y[\/latex]). These aren't mixed\u2014they differentiate with respect to the same variable twice.<\/p>","rendered":"

Higher-Order Partial Derivatives<\/h2>\n

Consider the function<\/p>\n

[latex]f(x,y)=2x^{3}-4xy^{2}+5y^{3}-6xy+5x-4y+12[\/latex].<\/p>\n

Its partial derivatives are<\/p>\n

[latex]\\dfrac{\\partial f}{\\partial x}=6x^{2}-4y^{2}-6y+5[\/latex] and [latex]\\dfrac{\\partial f}{\\partial y}=-8xy+15y^{2}-6x-4[\/latex].<\/p>\n

Each of these partial derivatives is itself a function of two variables, so we can calculate partial derivatives of these functions. Just as with single-variable derivatives, we call these second-order derivatives, third-order derivatives, and so on. In general, they are higher-order partial derivatives<\/strong>.<\/p>\n

There are four second-order partial derivatives for any function (provided they all exist):<\/p>\n

[latex]\\large{\\frac{\\partial^{2}f}{\\partial x^{2}}=\\frac{\\partial}{\\partial x}\\left[\\frac{\\partial f}{\\partial x}\\right],\\ \\frac{\\partial^{2}f}{\\partial x\\partial y}=\\frac{\\partial}{\\partial x}\\left[\\frac{\\partial f}{\\partial y}\\right],\\ \\frac{\\partial^{2}f}{\\partial y\\partial x}=\\frac{\\partial}{\\partial y}\\left[\\frac{\\partial f}{\\partial x}\\right],\\ \\frac{\\partial^{2}f}{\\partial y^{2}}=\\frac{\\partial}{\\partial y}\\left[\\frac{\\partial f}{\\partial y}\\right]}.[\/latex]<\/p>\n

An alternative notation for these is [latex]f_{xx}[\/latex], [latex]f_{yx}[\/latex], [latex]f_{xy}[\/latex], and [latex]f_{yy}[\/latex], respectively. Higher-order partial derivatives calculated with respect to different variables, such as [latex]f_{xy}[\/latex] and [latex]f_{yx}[\/latex], are called mixed partial derivatives<\/strong>.<\/p>\n

\n

In [latex]\\frac{\\partial^{2}f}{\\partial y\\partial x}[\/latex], read the variables from right to left: differentiate first with respect to [latex]x[\/latex], then with respect to [latex]y[\/latex].<\/p>\n

In subscript notation [latex]f_{xy}[\/latex], read left to right: differentiate first with respect to [latex]x[\/latex], then with respect to [latex]y[\/latex].<\/p>\n<\/section>\n

Calculate all four second partial derivates for the function<\/p>\n
[latex]\\large{f\\,(x,\\ y)=xe^{-3y}+\\sin{(2x-5y)}}[\/latex].<\/div>\n