{"id":169,"date":"2021-07-30T17:24:14","date_gmt":"2021-07-30T17:24:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&p=169"},"modified":"2022-10-29T02:05:20","modified_gmt":"2022-10-29T02:05:20","slug":"introduction-to-directional-derivatives-and-the-gradient","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/introduction-to-directional-derivatives-and-the-gradient\/","title":{"raw":"Introduction to Directional Derivatives and the Gradient","rendered":"Introduction to Directional Derivatives and the Gradient"},"content":{"raw":"In Partial Derivatives<\/em> we introduced the partial derivative. A function [latex]z=f(x,y)[\/latex] has two partial derivatives: [latex]\\partial{z} {\/} \\partial{x}[\/latex] and [latex]\\partial{z} {\/} \\partial{y}[\/latex]. These derivatives correspond to each of the independent variables and can be interpreted as instantaneous rates of change (that is, as slopes of a tangent line). For example, [latex]\\partial{z} {\/} \\partial{x}[\/latex] represents the slope of a tangent line passing through a given point on the surface defined by [latex]z=f(x,y)[\/latex], assuming the tangent line is parallel to the [latex]x[\/latex]-axis. Similarly, [latex]\\partial{z} {\/} \\partial{y}[\/latex] represents the slope of the tangent line parallel to the [latex]y[\/latex]-axis. Now we consider the possibility of a tangent line parallel to neither axis.","rendered":"

In Partial Derivatives<\/em> we introduced the partial derivative. A function [latex]z=f(x,y)[\/latex] has two partial derivatives: [latex]\\partial{z} {\/} \\partial{x}[\/latex] and [latex]\\partial{z} {\/} \\partial{y}[\/latex]. These derivatives correspond to each of the independent variables and can be interpreted as instantaneous rates of change (that is, as slopes of a tangent line). For example, [latex]\\partial{z} {\/} \\partial{x}[\/latex] represents the slope of a tangent line passing through a given point on the surface defined by [latex]z=f(x,y)[\/latex], assuming the tangent line is parallel to the [latex]x[\/latex]-axis. Similarly, [latex]\\partial{z} {\/} \\partial{y}[\/latex] represents the slope of the tangent line parallel to the [latex]y[\/latex]-axis. Now we consider the possibility of a tangent line parallel to neither axis.<\/p>\n\n\t\t\t

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