Successive Derivative of a Function

Successive differentiation: In scientific and engineering applications, higher-order differential coefficients are critical. Consider the function y=f(x) as a function of x. The derivative or first derivative of y with respect to x is defined as the result of differentiation y with respect to x, and it is symbolised by dy/dx. The outcome of differentiating f′(x) with respect to x is then defined as the function’s second derivative, denoted as d²y/dx².

What does the derivative of a function represent?

The term “successive differentiation” refers to a procedure that can be specified n times. The process for differentiating a given the function numerous times is known as successive differentiation, and the results are known as successive derivatives. Let us understand more about successive differentiation in this article.

The derivative or differential coefficient f'(x) of a function f of such a variable x defined in such a domain D has been discovered to be another function of x specified in a domain D’, which is a subset of D.

It’s possible that derived function f'(x) can now be differentiated with respect to variables x to provide another x function. Second order derivative of function f is the name given to this new function, indicated by f”(x). If we differentiate this function f”(x) again with regard to x, we get the third order derivative, indicated by f”‘ (x). The result of successively differentiating a function n time is fn(x), where n is any positive integer.

In the discussion on the expansion of functions in series and the formulation of differential equations, the use and importance of successive derivatives of a function will be realised.

A Function’s Derivative:

The rate where the value y of function varies with regard to the changing of the variable x is measured by the derivative of y=f(x). The derivative of the f(x) with regard to x is known as the derivative of f(x).

Let f(x) be just a function whose domain is the set of x values that have the following limit.

f′(x) =lim _h→0 f(x+h)–f(x)/h

Differences in Chains:

Allow y = x⁵ to be the case.

f′(x) = 5x⁴ for the first differentiation

F′′(x) = 54x³= 20x³ for the second differentiation

f′′′(x) = 543x² = 60x² for the third differentiation

fv(x) = 5432x = 120x for the fourth differentiation

fv(x) = 543221 = 120 is the fifth differentiation.

fvi(x) = 0 when it comes to sixth differentiation.

What is Successive Differentiation, and how does it work?

Successive differentiation is a process of deriving higher-order derivatives from a function by sequentially differentiating it.

1. If y = f(x) is a function of x, then dy/dx or dy or f′(x) or y1 is the derivative of y with respect to x. The first-order derivatives of y are this.
2. If dy/dx is differentiated again, y = f(x) is derivable double with respect to x, then d2y/dx² or d2y or f′′(x) or y² is the derivative of dy/dx with regard to x. The 2nd derivative of y is this.
3. If d²y/dx² is differentiated twice, y = f(x) is derivable three with respect to x, then d³y/dx³ or d³y or f′′′(x) or y³ is the derivative of d²y/dx² with respect to x.

The 3rd derivative of y is what it’s called.

Similarly, the successive derivatives may be found, and the nth derivative of y can be found by differentiating a given function n times with respect to x.

For the consecutive derivatives of y with respect to x, the following notations are commonly used.

The Theorem of Leibnitz:

Assume u(x) and v(x) are two functions with derivatives up to the nth order.

Let’s calculate the derivative of product of these 2 functions.

(uv)′=u′v +uv′ (uv)′=u′v + uv′ (uv)′=u′v + uv′ (uv)′=u′

If we differentiate again, we get (uv)′′=[(uv)′] ′ = (u′v +uv′) ′ 

                                                                                   =(u′v) ′+(uv′) ′ 

                                                                                   =u′′v + u′v′ + u′v′ +uv′′ 

                                                                                   =u′′v+2u′v′+uv′′ 

                                                                                   =u′′v+2u′v′+uv′′ 

                                                                                   =u′′v+2u′v′+uv′′ 

                                                                                   =u′′v+2u

Similarly, for the third derivative, we have (uv)′′′=[(uv)′′] ′ 

                                                                                      =(u′′v+2u′v′+uv′′) ′ 

                                                                                      =(u′′v) ′+(2u′v′) ′+(uv′′) ′ 

                                                                                      =(u′′v) ′+(2u′v′) ′+(uv′′) ′ 

                                                                                      =(u′′v) ′+(2u′v′) ′+(uv′′) ′ 

We can observe that the aforementioned formulas are quite similar to binomial expansion increased to the exponent by comparing them. We can obtain the formula for thenth order of derivative product of two functions by considering terms with zero powers, such as u0 and v0, which correspond to the functions u and v themselves.

(uv)ⁿ = _i=0 n (n i) u^(n–i) vⁱ, where (ni) is the number of combinations on the nth element.

The Leibnitz Rule is the name for this formula.

Conclusion:

If y = f(x) is a function of x, then dy/dx represents the derivative or differential coefficient of y with respect to x. If dy/dx can be differentiated again, that is, if y = f(x) can be derivable twice with regard to x, then d²y/dx² is the derivative of dy/dx with respect to x.

So successive differentiation is a process of successively determining the derivative of a function, and successive derivatives are the products of such differentiation. In the consecutive differentiations, we derive the higher-order derivatives using the conventional formula. As a result, d²y/dx², d³y/dx³, and so on are used to express higher-order derivatives. The value of higher-order derivatives in scientific & engineering applications cannot be overstated.