Name: partial derivatives and tangent planes
Uploaded: 2026-06-08T09:38:53.569Z
Duration: 1 min 14 s
Channel: JoVE

Imagine an irregular hill with varying elevation, where every point on the terrain is described using Cartesian coordinates. The terrain itself is described by a function z equals f of x, y, where z denotes the elevation corresponding to the point (x, y) in the plane.Now consider a specific point on the surface, where the goal is to understand how the height changes while moving in a particular direction.The partial derivative with respect to x measures how steep the terrain becomes when the movement is only along the x-axis, keeping the y-direction fixed.Likewise, the partial derivative with respect to y measures the steepness when the movement is only along the y-axis while keeping the x-direction fixed.A negative value of the partial derivative indicates downhill movement, whereas a positive value indicates uphill movement in that direction.Together, these slopes define a flat tangent plane that rests against the hill, describing the surface&#x2019;s total tilt at that exact point.

interpretations of partial derivatives

A surface defined by a function of two variables can be visualized as a vast, uneven terrain, where each point is identified using Cartesian coordinates. The elevation of the terrain at any point is determined by a function that assigns a height value to every pair of horizontal coordinates. This representation allows the surface to be studied in terms of how its height varies across different directions.At a specific point on this terrain, understanding how the height changes requires examining movement along particular directions. If motion is restricted along the x-direction while keeping the y-coordinate fixed, the resulting change in elevation describes the steepness in that direction. Similarly, movement along the y-direction, with the x-coordinate held constant, reveals the steepness in the perpendicular direction. These directional slopes indicate whether the terrain rises or falls: positive values correspond to upward movement, while negative values indicate a downward slope.Although these directional measures provide useful insights, they represent only individual slices of the surface. When considered together, they define a planar surface that closely approximates the terrain at the given point. This plane, known as the tangent plane, touches the surface at that point and reflects its overall tilt. It serves as a local linear approximation, capturing how the surface behaves in the immediate vicinity and providing a simplified representation of the terrain&#x2019;s geometry.

Interpretations of Partial Derivatives

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Education

JoVE Core

Calculus

حساب التفاضل والتكامل

Unit

وحدة

Partial Derivatives and Gradients

المشتقات الجزئية والتدرجات

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

تفسيرات المشتقات الجزئية

partial derivatives and tangent planes

Partial derivatives describe how a surface changes in one direction at a time. A surface from a function of two variables can be pictured like uneven terrain. Each point on that terrain is located with Cartesian coordinates, and the function gives the height at each pair of horizontal coordinates.

To study the surface at one point, you can move in a single direction while holding the other variable fixed. If you move along the x-direction and keep y constant, the change in height gives the slope in that direction. If you move along the y-direction and keep x constant, you get the slope in the perpendicular direction.

These directional slopes show whether the surface rises or falls. A positive value means the surface goes up. A negative value means it goes down. Each partial derivative gives only one slice of the surface, but together they show the local tilt.

When the two slopes are considered together, they define a tangent plane. A tangent plane is a flat surface that touches the curved surface at the chosen point. It provides a local linear approximation and shows how the surface behaves near that point.

Partial Derivatives and Tangent Planes

functions of two variables

Graphs of Functions in Three Dimensions

Functions of two variables use two inputs to determine one output. The output depends on both independent variables at the same time. They are often written as z = f(x, y), where each ordered pair in the domain gives one unique output value. The domain of a function of two variables is the region in the horizontal plane that contains the input pairs. Each point in this region matches one output value. That output can be shown as a vertical height above the plane. When all of the heights are...

graphs of two-variable functions

Weather Maps as Two-Variable Surfaces

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level curves and contour maps

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functions of three or more variables

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limits of multivariable functions

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properties of limits in multivariable calculus

properties-of-limits-in-multivariable-calculus

Limit Laws for Multivariable Functions

Limit laws in multivariable calculus give clear rules for finding limits of functions with more than one variable. These rules help break a complex expression into smaller parts whose limits are already known. They are used when [1m\lim_{(x,y)\to(a,b)} f(x,y)=L[0m and [1m\lim_{(x,y)\to(a,b)} g(x,y)=M[0m exist. The sum law says the limit of a sum equals the sum of the limits. The difference law says the limit of a difference equals the difference of the limits. In notation,...

continuity for functions of multiple variables

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Testing Continuity with Paths in Two Variables

Continuity in multivariable functions describes how a function with two or more input variables changes near a point. It extends the idea of continuity from single-variable calculus into higher dimensions. This idea is important for modeling real-world situations that vary across space. A multivariable function is continuous at a point only if three conditions are met at the same time. The function must be defined at that point. The limit must exist as the input moves toward the point from any...

introduction to partial derivatives

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Partial derivatives help describe how a multivariable function changes when one input changes and the others stay fixed. In many real-world situations, one output depends on more than one input. A high-tech assembly plant is one example. Total production can depend on technician labor and machine capacity at the same time. This relationship can be written as a continuous function P(T, M), where T stands for technician labor input and M stands for machine capacity. When demand rises but the...

Partial derivatives describe how a surface changes in one direction at a time. A surface from a function of two variables can be pictured like uneven terrain. Each point on that terrain is located with Cartesian coordinates, and the function gives the height at each pair of horizontal coordinates. To study the surface at one point, you can move in a single direction while holding the other variable fixed. If you move along the x-direction and keep y constant, the change in height gives the...

multivariable functions and higher derivatives

multivariable-functions-and-higher-derivatives

Partial Derivatives and Surface Curvature

Partial derivatives and surface curvature help describe multivariable functions. A multivariable function gives one output for each ordered set of inputs. For a function f(x, y), each point (x, y) can be treated as a height z = f(x, y). This creates a surface in three-dimensional space. These functions often appear in physical models and optimization problems, where behavior depends on several changing variables. To study how the function changes in one direction, one variable is held fixed...

partial differential equations

Modeling Waves with Space and Time

Partial differential equations help describe waves that change across both space and time. A stone dropped into a still pond creates circular waves that spread outward. The water surface rises and falls at each point as the wave passes, while the shape of the surface also changes across the pond at any fixed moment. This kind of motion needs a mathematical model that tracks more than one variable at once. At one location, the water level changes as time moves forward. That change includes the...

tangent planes to surfaces

Using Partial Derivatives for Tangent Planes

Partial derivatives help describe a tangent plane for a surface in multivariable calculus. For a surface written as z = f(x, y), the tangent plane at one point gives the best linear approximation near that point. This makes a curved surface easier to study with a simple planar model. To build the tangent plane, take vertical slices through the surface. Hold x constant in one slice and y constant in another slice. Each slice forms a curve on the surface. The tangent lines to those curves at the...

linear approximations

Tangent Plane Estimates for Two Variables

Tangent plane estimates help approximate a differentiable function of two variables near a known point. Instead of using the curved surface directly, the function is replaced by its tangent plane. This gives a fast estimate for values close to the point being studied. For the function f(x,y)=x^2+3y^2, the point (2, 1) is used as the reference. The exact value there is f(2,1)=2^2+3(1)^2=4+3=7. The linear approximation near (a,b) is L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b). To build the...

multivariable chain rule

Total Derivatives in Connected Variables

Total derivatives in connected variables show how one change can affect several related quantities. Here, z depends on two intermediate variables, x and y, and both x and y change with a third variable, t. The connection is indirect, because t does not appear in the formula for z itself. The goal is to find the total derivative of z with respect to t, written as dz/dt. When the functions are differentiable, the change in z can be split into two parts. One part comes from how z changes with x...

implicit differentiation with partial derivatives

implicit-differentiation-with-partial-derivatives

Finding Tangent Slopes from Implicit Equations

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directional derivatives

Directional Derivatives in Any Direction

Directional derivatives measure how a surface changes in any chosen direction. In multivariable calculus, partial derivatives show change along one coordinate at a time. For a function of two variables, one partial derivative gives the slope in the x-direction, and the other gives the slope in the y-direction. These single-direction slopes are useful for local behavior. However, many real movements do not follow the coordinate axes. Fluid flow, heat transfer, and motion across uneven terrain...

gradient vectors and their applications

Steepest Ascent and the Gradient Vector

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maximizing the directional derivative

Gradient Direction and Rate of Change

The directional derivative shows how a multivariable function changes at a point when you move in a chosen direction. That direction is written as a unit vector, so only the direction matters and not the length of the vector. Changing the direction changes the rate of change, which is why the directional derivative depends strongly on the path you choose. The directional derivative is found using the gradient vector and a unit direction vector. The gradient tells you both the direction and the...

tangent planes to level surfaces

Gradient Vectors and Tangent Planes

Gradient vectors and tangent planes help describe the shape of a level surface at a point. A level surface is the set of all points in space where a function of three variables has the same fixed value. To understand the local geometry near one point, it is not enough to know the coordinates alone. You also need to know how the surface tilts there. One way to study that tilt is to follow a path that stays on the level surface and passes through the point. This path can be written as a...

significance of the gradient vector

Gradient Vector and Direction of Steepest Rise

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local maximum and minimum values

Finding Critical Points in Multivariable Calculus

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lagrange multipliers: one constraint

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lagrange multipliers: two constraints

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lagrange multipliers: problem solving

Optimizing Silo Design with Lagrange Multipliers

A silo with a cylindrical base and a hemispherical roof can be sized with Lagrange multipliers. This calculus method helps find the best design when the surface area must stay fixed. In this problem, the goal is to maximize storage volume while using a set amount of material. The model uses two main variables: the radius r of the base and the height h of the cylindrical section. The total volume comes from adding the volume of the cylinder and the hemisphere. The fixed surface area includes...

key terms and concepts

Learn how partial derivatives show the slope of a surface in the x and y directions, and how they combine to form a tangent plane at a point.

Partial Derivatives and Tangent Planes | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Partial Derivatives and Tangent Planes

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