Derivative vs Differential
In differential calculus, derivative and differential of a function are closely related but have very different meanings, and used to represent two important mathematical objects related to differentiable functions.
What is derivative?
Derivative of a function measures the rate at which the function value changes as its input changes. In multi-variable functions, the change in the function value depends on the direction of the change of the values of the independent variables. Therefore, in such cases, a specific direction is chosen and the function is differentiated in that particular direction. That derivative is called the directional derivative. Partial derivatives are a special kind of directional derivatives.
Derivative of a vector-valued function f can be defined as the limit wherever it exists finitely. As mentioned before, this gives us the rate of increase of the function f along the direction of the vector u. In the case of a single-valued function, this reduces to the well-known definition of the derivative,
For example, is everywhere differentiable, and the derivative is equal to the limit, , which is equal to . The derivatives of functions such as exist everywhere. They are respectively equal to the functions .
This is known as the first derivative. Usually the first derivative of function f is denoted by f (1). Now using this notation, it is possible to define higher order derivatives. is the second order directional derivative, and denoting the nth derivative by f (n) for each n, , defines the nth derivative.
What is differential?
Differential of a function represents the change in the function with respect to changes in the independent variable or variables. In the usual notation, for a given function f of a single variable x, the total differential of order 1 df is given by, . This means that for an infinitesimal change in x(i.e. dx), there will be a f (1)(x)dx change in f.
Using limits one can end up with this definition as follows. Assume ∆x is the change in x at an arbitrary point x and ∆f is the corresponding change in the function f. It can be shown that ∆f = f (1)(x)∆x+ ϵ, where ϵ is the error. Now, the limit ∆x→0∆f/∆x= f (1)(x) (using the previously stated definition of derivative) and thus, ∆x→0ϵ/∆x= 0. Therefore, it is possible to conclude that, ∆x→0ϵ = 0. Now, denoting ∆x→0 ∆f as df and ∆x→0 ∆x as dx the definition of the differential is rigorously obtained.
For example, the differential of the function is .
In the case of functions of two or more variables, the total differential of a function is defined as the sum of differentials in the directions of each of the independent variables. Mathematically, it can be stated as .
What is the difference between derivative and differential?
• Derivative refers to a rate of change of a function whereas the differential refers to the actual change of the function, when the independent variable is subjected to change.
• The derivative is given by , but the differential is given by .