Tangent Plane Notes

Published on: September 29, 2026

This Tangent Plane Calculator finds the plane that touches a surface at one point and shows the working line by line. The two partial derivatives at that point give the slopes along each axis, and those slopes become the coefficients of the plane. Close to the point the plane and the surface are nearly the same, which is what makes it a linear approximation. Following the same pattern every time keeps the point and the slopes separate.

Step-by-step method

Set up the problem.
Find the partial derivatives f_x and f_y.
Evaluate f, f_x, and f_y at the point.
Substitute into the tangent-plane formula.
Simplify the plane equation.

Formula:

\(z = f(a,b) + f_{x}(a,b)(x-a) + f_{y}(a,b)(y-b)\)

See Example 1 Hide Example 1

Problem

\(f = x^{2} + y^{2},\; P = (1, 2)\)

Approach

Step-by-step method

Set up the problem.
Find the partial derivatives f_x and f_y.
Evaluate f, f_x, and f_y at the point.
Substitute into the tangent-plane formula.
Simplify the plane equation.

Step 1

Step 1 - Set up the problem.

In this problem:

\(f = x^{2} + y^{2},\; P = (1, 2)\)

Step 2

Step 2 - Find the partial derivatives f_x and f_y.

In this problem:

\(f_x = 2 x,\quad f_y = 2 y\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x^{2}\), \(y^{2}\).

\(f\left(x\right) = x^{2} + y^{2}\)

Step 2

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dx}\left(f\left(x\right) + g\left(x\right)\right) = \frac{d}{dx}f\left(x\right) + \frac{d}{dx}g\left(x\right)\)

Step 3

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dx}\left(x^{2} + y^{2}\right) = \frac{d}{dx}\left(x^{2}\right) + \frac{d}{dx}\left(y^{2}\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x^{2}\).

In this problem: The separated term is \(x^{2}\). Use the power rule to get its derivative.

\(\frac{d}{dx}\left(x^{2}\right) = 2 x\)

Problem

\(f\left(x\right) = {x}^{2}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x^{2}\). This is a power of \(x\), so the exponent is \(n = 2\).

\(f\left(x\right) = {x}^{2}\)

Step 2

Step 2 - Write the power rule formula.

In this problem: The power rule says to move the exponent to the front, then subtract 1 from the exponent.

\(\frac{d}{dx}{x}^{n} = n\,{x}^{n - 1}\)

Step 3

Step 3 - Substitute the given power into the formula.

In this problem: Here, \(n = 2\). Substitute this exponent into the power rule, but do not simplify yet.

\(\frac{d}{dx}{x}^{2} = 2{x}^{2 - 1}\)

Step 4

Step 4 - Solve and simplify.

In this problem: Now subtract 1 from the exponent and keep the result in power form. This gives \(2{x}^{1}\).

\(2{x}^{2 - 1} = 2{x}^{1} = 2 x\)

Final Answer

\(2 x\)

Step 4b

Step 4b - Differentiate the separated term \(y^{2}\).

In this problem: The separated term is \(y^{2}\). Use the constant rule to get its derivative.

\(\frac{d}{dx}\left(y^{2}\right) = 0\)

Problem

\(f\left(x\right) = y^{2}\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(y^{2}\). This expression does not contain \(x\), so it is a constant.

\(f\left(x\right) = y^{2}\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(y^{2}\) is constant, its derivative is \(0\).

\(\frac{d}{dx}\left(y^{2}\right) = 0\)

Final Answer

\(0\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(x\right) = 2 x + 0 = 2 x\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x^{2}\), \(y^{2}\).

\(f\left(y\right) = x^{2} + y^{2}\)

Step 2

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dy}\left(f\left(y\right) + g\left(y\right)\right) = \frac{d}{dy}f\left(y\right) + \frac{d}{dy}g\left(y\right)\)

Step 3

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dy}\left(x^{2} + y^{2}\right) = \frac{d}{dy}\left(x^{2}\right) + \frac{d}{dy}\left(y^{2}\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x^{2}\).

In this problem: The separated term is \(x^{2}\). Use the constant rule to get its derivative.

\(\frac{d}{dy}\left(x^{2}\right) = 0\)

Problem

\(f\left(y\right) = x^{2}\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x^{2}\). This expression does not contain \(y\), so it is a constant.

\(f\left(y\right) = x^{2}\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(x^{2}\) is constant, its derivative is \(0\).

\(\frac{d}{dy}\left(x^{2}\right) = 0\)

Final Answer

\(0\)

Step 4b

Step 4b - Differentiate the separated term \(y^{2}\).

In this problem: The separated term is \(y^{2}\). Use the power rule to get its derivative.

\(\frac{d}{dy}\left(y^{2}\right) = 2 y\)

Problem

\(f\left(y\right) = {y}^{2}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(y^{2}\). This is a power of \(x\), so the exponent is \(n = 2\).

\(f\left(y\right) = {y}^{2}\)

Step 2

Step 2 - Write the power rule formula.

In this problem: The power rule says to move the exponent to the front, then subtract 1 from the exponent.

\(\frac{d}{dy}{y}^{n} = n\,{y}^{n - 1}\)

Step 3

Step 3 - Substitute the given power into the formula.

In this problem: Here, \(n = 2\). Substitute this exponent into the power rule, but do not simplify yet.

\(\frac{d}{dy}{y}^{2} = 2{y}^{2 - 1}\)

Step 4

Step 4 - Solve and simplify.

In this problem: Now subtract 1 from the exponent and keep the result in power form. This gives \(2{y}^{1}\).

\(2{y}^{2 - 1} = 2{y}^{1} = 2 y\)

Final Answer

\(2 y\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(y\right) = 0 + 2 y = 2 y\)

Step 3

Step 3 - Evaluate f, f_x, and f_y at the point.

In this problem:

\(f(1,2) = 5,\quad f_x = 2,\quad f_y = 4\)

Step 4

Step 4 - Substitute into the tangent-plane formula.

In this problem:

\(z = 5 + 2\left(x - 1\right) + 4\left(y - 2\right)\)

Step 5

Step 5 - Simplify the plane equation.

In this problem:

\(z = 2 x + 4 y - 5\)

Final Answer

\(z = 2 x + 4 y - 5\)

See Example 2 Hide Example 2

Problem

\(f = x y,\; P = (2, 3)\)

Approach

Step-by-step method

Set up the problem.
Find the partial derivatives f_x and f_y.
Evaluate f, f_x, and f_y at the point.
Substitute into the tangent-plane formula.
Simplify the plane equation.

Step 1

Step 1 - Set up the problem.

In this problem:

\(f = x y,\; P = (2, 3)\)

Step 2

Step 2 - Find the partial derivatives f_x and f_y.

In this problem:

\(f_x = y,\quad f_y = x\)

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(x y\). The constant coefficient is \(y\), and the variable part is \(x\).

\(x y = y\cdot \left(x\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dx}\left(c\cdot f\left(x\right)\right) = c\cdot \frac{d}{dx}f\left(x\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(y\) outside the derivative and leave \(x\) inside the derivative.

\(\frac{d}{dx}\left(x y\right) = y\cdot \frac{d}{dx}\left(x\right)\)

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(x y\). The constant coefficient is \(x\), and the variable part is \(y\).

\(x y = x\cdot \left(y\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dy}\left(c\cdot f\left(y\right)\right) = c\cdot \frac{d}{dy}f\left(y\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(x\) outside the derivative and leave \(y\) inside the derivative.

\(\frac{d}{dy}\left(x y\right) = x\cdot \frac{d}{dy}\left(y\right)\)

Step 3

Step 3 - Evaluate f, f_x, and f_y at the point.

In this problem:

\(f(2,3) = 6,\quad f_x = 3,\quad f_y = 2\)

Step 4

Step 4 - Substitute into the tangent-plane formula.

In this problem:

\(z = 6 + 3\left(x - 2\right) + 2\left(y - 3\right)\)

Step 5

Step 5 - Simplify the plane equation.

In this problem:

\(z = 3 x + 2 y - 6\)

Final Answer

\(z = 3 x + 2 y - 6\)

See Example 3 Hide Example 3

Problem

\(f = x^{2} y,\; P = (1, 3)\)

Approach

Step-by-step method

Set up the problem.
Find the partial derivatives f_x and f_y.
Evaluate f, f_x, and f_y at the point.
Substitute into the tangent-plane formula.
Simplify the plane equation.

Step 1

Step 1 - Set up the problem.

In this problem:

\(f = x^{2} y,\; P = (1, 3)\)

Step 2

Step 2 - Find the partial derivatives f_x and f_y.

In this problem:

\(f_x = 2 x y,\quad f_y = x^{2}\)

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(x^{2} y\). The constant coefficient is \(y\), and the variable part is \(x^{2}\).

\(x^{2} y = y\cdot \left(x^{2}\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dx}\left(c\cdot f\left(x\right)\right) = c\cdot \frac{d}{dx}f\left(x\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(y\) outside the derivative and leave \(x^{2}\) inside the derivative.

\(\frac{d}{dx}\left(x^{2} y\right) = y\cdot \frac{d}{dx}\left(x^{2}\right)\)

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(x^{2} y\). The constant coefficient is \(x^{2}\), and the variable part is \(y\).

\(x^{2} y = x^{2}\cdot \left(y\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dy}\left(c\cdot f\left(y\right)\right) = c\cdot \frac{d}{dy}f\left(y\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(x^{2}\) outside the derivative and leave \(y\) inside the derivative.

\(\frac{d}{dy}\left(x^{2} y\right) = x^{2}\cdot \frac{d}{dy}\left(y\right)\)

Step 3

Step 3 - Evaluate f, f_x, and f_y at the point.

In this problem:

\(f(1,3) = 3,\quad f_x = 6,\quad f_y = 1\)

Step 4

Step 4 - Substitute into the tangent-plane formula.

In this problem:

\(z = 3 + 6\left(x - 1\right) + 1\left(y - 3\right)\)

Step 5

Step 5 - Simplify the plane equation.

In this problem:

\(z = 6 x + y - 6\)

Final Answer

\(z = 6 x + y - 6\)

See Example 4 Hide Example 4

Problem

\(f = x^{2} + 2 y^{2},\; P = (2, 1)\)

Approach

Step-by-step method

Set up the problem.
Find the partial derivatives f_x and f_y.
Evaluate f, f_x, and f_y at the point.
Substitute into the tangent-plane formula.
Simplify the plane equation.

Step 1

Step 1 - Set up the problem.

In this problem:

\(f = x^{2} + 2 y^{2},\; P = (2, 1)\)

Step 2

Step 2 - Find the partial derivatives f_x and f_y.

In this problem:

\(f_x = 2 x,\quad f_y = 4 y\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x^{2}\), \(2 y^{2}\).

\(f\left(x\right) = x^{2} + 2 y^{2}\)

Step 2

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dx}\left(f\left(x\right) + g\left(x\right)\right) = \frac{d}{dx}f\left(x\right) + \frac{d}{dx}g\left(x\right)\)

Step 3

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dx}\left(x^{2} + 2 y^{2}\right) = \frac{d}{dx}\left(x^{2}\right) + \frac{d}{dx}\left(2 y^{2}\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x^{2}\).

In this problem: The separated term is \(x^{2}\). Use the power rule to get its derivative.

\(\frac{d}{dx}\left(x^{2}\right) = 2 x\)

Problem

\(f\left(x\right) = {x}^{2}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x^{2}\). This is a power of \(x\), so the exponent is \(n = 2\).

\(f\left(x\right) = {x}^{2}\)

Step 2

Step 2 - Write the power rule formula.

In this problem: The power rule says to move the exponent to the front, then subtract 1 from the exponent.

\(\frac{d}{dx}{x}^{n} = n\,{x}^{n - 1}\)

Step 3

Step 3 - Substitute the given power into the formula.

In this problem: Here, \(n = 2\). Substitute this exponent into the power rule, but do not simplify yet.

\(\frac{d}{dx}{x}^{2} = 2{x}^{2 - 1}\)

Step 4

Step 4 - Solve and simplify.

In this problem: Now subtract 1 from the exponent and keep the result in power form. This gives \(2{x}^{1}\).

\(2{x}^{2 - 1} = 2{x}^{1} = 2 x\)

Final Answer

\(2 x\)

Step 4b

Step 4b - Differentiate the separated term \(2 y^{2}\).

In this problem: The separated term is \(2 y^{2}\). Use the constant rule to get its derivative.

\(\frac{d}{dx}\left(2 y^{2}\right) = 0\)

Problem

\(f\left(x\right) = 2 y^{2}\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(2 y^{2}\). This expression does not contain \(x\), so it is a constant.

\(f\left(x\right) = 2 y^{2}\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(2 y^{2}\) is constant, its derivative is \(0\).

\(\frac{d}{dx}\left(2 y^{2}\right) = 0\)

Final Answer

\(0\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(x\right) = 2 x + 0 = 2 x\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x^{2}\), \(2 y^{2}\).

\(f\left(y\right) = x^{2} + 2 y^{2}\)

Step 2

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dy}\left(f\left(y\right) + g\left(y\right)\right) = \frac{d}{dy}f\left(y\right) + \frac{d}{dy}g\left(y\right)\)

Step 3

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dy}\left(x^{2} + 2 y^{2}\right) = \frac{d}{dy}\left(x^{2}\right) + \frac{d}{dy}\left(2 y^{2}\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x^{2}\).

In this problem: The separated term is \(x^{2}\). Use the constant rule to get its derivative.

\(\frac{d}{dy}\left(x^{2}\right) = 0\)

Problem

\(f\left(y\right) = x^{2}\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x^{2}\). This expression does not contain \(y\), so it is a constant.

\(f\left(y\right) = x^{2}\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(x^{2}\) is constant, its derivative is \(0\).

\(\frac{d}{dy}\left(x^{2}\right) = 0\)

Final Answer

\(0\)

Step 4b

Step 4b - Differentiate the separated term \(2 y^{2}\).

In this problem: The separated term is \(2 y^{2}\). Use the power rule to get its derivative.

\(\frac{d}{dy}\left(2 y^{2}\right) = 4 y\)

Problem

\(f\left(y\right) = 2{y}^{2}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(2 y^{2}\). This is a constant multiple of a power of \(x\). The coefficient is \(2\), the power part is \(y^{2}\), and the exponent is \(n = 2\).

\(f\left(y\right) = 2{y}^{2}\)

Step 2

Step 2 - Apply the constant multiple rule first.

In this problem: Before using the power rule, apply the constant multiple rule because \(2\) is a constant coefficient.

\(\frac{d}{dy}2{y}^{2} = 2\cdot \frac{d}{dy}{y}^{2}\)

Problem

\(f\left(y\right) = 2 y^{2}\)

Approach

Step-by-step method

Set up the problem.
Write the constant multiple rule formula.
Apply the constant multiple rule.

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(2 y^{2}\). The constant coefficient is \(2\), and the variable part is \(y^{2}\).

\(2 y^{2} = 2\cdot \left(y^{2}\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dy}\left(c\cdot f\left(y\right)\right) = c\cdot \frac{d}{dy}f\left(y\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(2\) outside the derivative and leave \(y^{2}\) inside the derivative.

\(\frac{d}{dy}\left(2 y^{2}\right) = 2\cdot \frac{d}{dy}\left(y^{2}\right)\)

Final Answer

\(2\cdot \frac{d}{dy}\left(y^{2}\right)\)

Step 3

Step 3 - Write the power rule formula.

In this problem: Now use the power rule on the remaining power of \(x\).

\(\frac{d}{dy}{y}^{n} = n\,{y}^{n - 1}\)

Step 4

Step 4 - Substitute the exponent into the formula.

In this problem: Here, \(n = 2\). Substitute this exponent into the power rule, then keep the coefficient \(2\) outside.

\(2\cdot \frac{d}{dy}{y}^{2} = 2\cdot 2{y}^{2 - 1}\)

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(y^{2}\), multiply by the constant \(2\), and keep the solving line in power form. This gives \(4{y}^{1}\).

\(2\cdot 2{y}^{1} = 4{y}^{1} = 4 y\)

Final Answer

\(4 y\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(y\right) = 0 + 4 y = 4 y\)

Step 3

Step 3 - Evaluate f, f_x, and f_y at the point.

In this problem:

\(f(2,1) = 6,\quad f_x = 4,\quad f_y = 4\)

Step 4

Step 4 - Substitute into the tangent-plane formula.

In this problem:

\(z = 6 + 4\left(x - 2\right) + 4\left(y - 1\right)\)

Step 5

Step 5 - Simplify the plane equation.

In this problem:

\(z = 4 x + 4 y - 6\)

Final Answer

\(z = 4 x + 4 y - 6\)