Tangent Plane Calculator

Published on: July 21, 2026

Final Answer: Free • Full Steps: Pro

This Tangent Plane Calculator finds the plane tangent to the surface \(z = f(x, y)\) at a point (the linear approximation). It computes the partial derivatives, evaluates them at the point, and assembles the plane equation step by step.

Step-by-step method

Set up the surface and the point.
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:

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

Example 1:

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

Step 1 - Set up the surface and the point.

In this problem: We build the tangent plane to \(f = x^{2} + y^{2}\) at \((1, 2)\).

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

Step 2 - Find the partial derivatives f_x and f_y.

In this problem: The partials are \(f_x = 2 x\) and \(f_y = 2 y\).

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

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

In this problem: At the point: \(f = 5\), \(f_x = 2\), \(f_y = 4\).

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

Step 4 - Substitute into the tangent-plane formula.

In this problem: Substitute these values into the tangent-plane formula.

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

Step 5 - Simplify the plane equation.

In this problem: Simplifying gives \(z = 2 x + 4 y - 5\).

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

Final answer:

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

Example 2:

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

Step 1 - Set up the surface and the point.

In this problem: We build the tangent plane to \(f = x y\) at \((2, 3)\).

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

Step 2 - Find the partial derivatives f_x and f_y.

In this problem: The partials are \(f_x = y\) and \(f_y = x\).

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

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

In this problem: At the point: \(f = 6\), \(f_x = 3\), \(f_y = 2\).

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

Step 4 - Substitute into the tangent-plane formula.

In this problem: Substitute these values into the tangent-plane formula.

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

Step 5 - Simplify the plane equation.

In this problem: Simplifying gives \(z = 3 x + 2 y - 6\).

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

Final answer:

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