Tangent Plane — Definition, Formula & Examples

A tangent plane is the flat surface that just touches a smooth surface at a given point and best approximates the surface near that point. It generalizes the idea of a tangent line to functions of two variables.

Given a surface defined by $z = f(x, y)$ where $f$ has continuous partial derivatives at a point $(a, b)$ , the tangent plane at $(a, b, f(a,b))$ is the plane whose equation is $z - f(a,b) = f_x(a,b)(x - a) + f_y(a,b)(y - b)$ , where $f_x$ and $f_y$ denote the partial derivatives of $f$ with respect to $x$ and $y$ .

Key Formula

z = f(a,b) + f_x(a,b)(x - a) + f_y(a,b)(y - b)

Where:

$f(a,b)$ = Value of the function at the point of tangency
$f_x(a,b)$ = Partial derivative of f with respect to x, evaluated at (a, b)
$f_y(a,b)$ = Partial derivative of f with respect to y, evaluated at (a, b)
$(a, b)$ = The point in the xy-plane where the tangent plane is computed

How It Works

To find the tangent plane at a point on a surface

z = f(x,y)

, you compute the two partial derivatives

f_x

and

f_y

, evaluate them at the point of tangency, and substitute into the tangent plane formula. The partial derivatives act as slopes:

f_x(a,b)

gives the slope in the

x

-direction and

f_y(a,b)

gives the slope in the

y

-direction. Together they determine the orientation of the plane. The tangent plane is also the foundation for linear approximation in two variables, since near

(a,b)

the surface is well-approximated by this plane.

Worked Example

Problem: Find the equation of the tangent plane to the surface z = x² + y² at the point (1, 2, 5).

Step 1: Compute the partial derivatives of f(x, y) = x² + y².

f_x = 2x, \quad f_y = 2y

Step 2: Evaluate the partial derivatives at the point (1, 2).

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

Step 3: Substitute into the tangent plane formula with a = 1, b = 2, and f(1,2) = 5.

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

Answer: The tangent plane is

z = 2x + 4y - 5

Why It Matters

Tangent planes are essential for linear approximation of multivariable functions, which appears throughout engineering and physics when simplifying complex surfaces locally. They also provide the geometric basis for understanding the gradient vector, since the gradient at a point is perpendicular to the level curve and determines the tangent plane's tilt.

Common Mistakes

Mistake: Forgetting to evaluate the partial derivatives at the specific point before substituting into the formula.

Correction: Always plug in the coordinates (a, b) into f_x and f_y first. The tangent plane uses constant slopes, not general expressions.