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 multivariable 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 vectorvalued 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 singlevalued function, this reduces to the wellknown 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^{(n1)}(x+h)f^{(n1)}(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].
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 [latex]\\frac{df}{dx}=\\lim_{h \to 0}\\frac{f(x+h)f(x)}{h}[/latex], but the differential is given by [latex]df = f^{1}(x)dx[/latex].

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