Partial differentiation: Tangent planes

The Concept of Partial Differentiation

In mathematics, partial differentiation is a process to find the rate of change of a function with respect to one of its variables, while keeping the other variables constant.
It is a key concept in multivariable calculus and is used when dealing with functions of more than one variable.
Partial differentiation is represented as ∂f/∂x or ∂f/∂y, where ∂ is the partial derivative symbol.
This process is called ‘partial’ as it only considers a ‘part’ of the variables, treating others as constants.

A tangent plane is a geometric plane which touches a surface at only one point, called a point of tangency.
In multivariable calculus, the concept of a tangent line extends to the idea of a tangent plane for surfaces in three dimensions.
The tangent plane is an important concept when studying surfaces and it approximates the surface near a given point.

The idea of partial differentiation and tangent planes are interconnected in multivariable calculus. This relationship is what enables us to accurately model the behavior and properties of three-dimensional objects and surfaces.
Given a function z = f(x, y), the tangent plane to the surface at a point can be found using the values of the partial derivatives of f with respect to x and y.
The equation of the tangent plane at a given point (a, b, f(a, b)) for function z=f(x, y) is given by: z - f(a, b) = f_x(a, b)(x - a) + f_y(a, b)(y - b), where f_x and f_y are the partial derivatives of the function at point (a, b).
To obtain the tangent plane, you must first calculate the partial derivatives, substitute the point of interest into these derivatives, and then substitute all these values into the equation of the tangent plane.

Partial differentiation and tangent planes play a key role in many branches of mathematics and its applications, including physics, engineering, and economics.
It is used to find the local approximation of a function, which is useful in various analyses and calculations.
By learning these two elements, you will be able to solve a wide range of mathematical problems related to rates of change and approximations in three dimensions.