Difference Between Differentiation and Derivative

Differentiation vs Derivative
 

In differential calculus, derivative and differentiation are closely related, but very different, and used to represent two important mathematical concepts related to 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 differentiation?

Differentiation is the process of finding the derivative of a differentiable function. D-operator denoted by D represents differentiation in some contexts. If x is the independent variable, then D ≡ ^d/_dx. The D-operator is a linear operator, i.e. for any two differentiable function f and g and constant c, following properties hold.

I.  D(f + g) = D(f) + D(g)

II.  D(cf) = cD(f )

Using the D-operator, the other rules associated with differentiation can be expressed as follows. D(f g) = D(f ) g +f D(g) , D(^f/_g) = ^{[D(f ) g – f D(g)]}/_g² and D(f  o g) = (D(f) o g) D(g).

For example, when F(x) = x²sin x is differentiated with respect to x using the rules given, the answer will be 2xsin x ­+ x²cosx.