Difference Between Derivative and Differential

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 [latex]\\frac{df}{d\\boldsymbol{u}}=\\lim_{h \to 0}\\frac{f(\\boldsymbol{x}+h \\boldsymbol{u})-f(\\boldsymbol{x})}{h}[/latex] 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,  [latex]\\frac{df}{dx}=\\lim_{h \\to 0}\\frac{f(x+h)-f(x)}{h}[/latex]

For example, [latex]f(x)=x^{3}+4x+5[/latex] is everywhere differentiable, and the derivative is equal to the limit, [latex]\\lim_{h \\to 0}\\frac{(x+h)^{3}+4(x+h)+5-(x^{3}+4x+5)}{h}[/latex], which is equal to [latex]3x^{2}+4[/latex]. The derivatives of functions such as  [latex]e^{x}, \\sin x, \\cos x[/latex] exist everywhere. They are respectively equal to the functions [latex]e^{x}, \\cos x, – \\sin x[/latex].                                                                                

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. [latex]\\frac{d^{2}f}{dx^{2}}=\\lim_{h \\to 0}\\frac{f^{(1)}(x+h)-f^{(1)}(x)}{h}[/latex] is the second order directional derivative, and denoting the n^th derivative by f ⁽ⁿ⁾ for each n, [latex]\\frac{d^{n}f}{dx^{n}}=\\lim_{h \\to 0}\\frac{f^{(n-1)}(x+h)-f^{(n-1)}(x)}{h}[/latex],  defines the n^th 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, [latex]df = f^{1}(x)dx[/latex]. 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→0^ϵ/_∆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 [latex]f(x)=x^{3}+4x+5[/latex] is [latex](3x^{2}+4)dx[/latex].

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 [latex]df = \\sum_{i=1}^{n} \\frac{\\partial f}{\\partial x_{i}}dx_{i}[/latex].