Successive Derivatives of a Function

It is possible to calculate the derivative of a function of a real variable in mathematics by measuring the sensitivity of the function value (output value) to changes in its argument (input value). Calculus’s derivatives are a fundamental tool for problem-solving. A good example is the derivative of the position of a moving object with respect to time, which is also known as the object’s velocity: it measures how quickly the position of the object changes as time progresses.

When a derivative of a single variable exists at a given input value, the slope of the tangent line to the graph of the function at that point is equal to the slope of the derivative of the single variable at that point. The tangent line is the best linear approximation of the function near the input value, and it is drawn through the origin. As a result, the derivative is frequently referred to as the “instantaneous rate of change,” which is defined as the ratio of the instantaneous change in the dependent variable to the instantaneous change in the independent variable.

Differentiation is the term used to describe the process of determining a derivative. Antidifferentiation is the term used to describe the opposite process. The fundamental theorem of calculus establishes a connection between antidifferentiation and integration. Differentiation and integration are the two fundamental operations in single-variable calculus, and they are also the most commonly used.

What Is Successive Differentiation? 

Successive differentiation is the process of deriving higher-order derivatives of a function by differentiating the function successively.
 

• In this case, the derivative of y with respect to x is denoted by dy/dx. This is the first-order derivative of y. 

• Assuming that the function y=f(x) is derivable twice with respect to x and that dy/dx is differentiated again, the derivative of dy/dx with respect to x is denoted by either d2y/dx2 or D2y or f′′(x) or y2. This is the derivative of y in the second order.

• If d2y/dx2 is differentiated again, and y=f(x) is derivable thrice with respect to x, then the derivative of d2y/dx2 with respect to x is symbolised by d3y/dx3 or D3y or f′′′(x) or y3. If d2y/dx2 is differentiated again, and y=f(x) is derivable thrice with respect to x, then This is referred to as the third-order derivative of the variable y.

To find the consecutive derivatives, we must differentiate the given function n times with respect to x. In general, we can find the nth derivative of y by differentiating it n times with respect to x.

In scientific and engineering applications, higher-order differential coefficients are of critical importance because they allow for more accurate calculations. Let y=f(x) be a function of the variable x. If y is differentiable with regard to x, the result of differentiating y from x is defined as the derivative or the first derivative of y with respect to x, and it is symbolised by the symbol dy/dx.

When we differentiate f′(x) with respect to x, we get the second derivative of the function, which is denoted by the symbols d2y/dx2, which stands for the second derivative of the function.

This process, which is known as successive differentiation, can be defined to occur n times. Differentiating a given function several times is known as successive differentiation; the derivatives obtained as a result are referred to as successive derivatives.

Example

Que: Find y2 for the following function y=e3x+2.

Solution: We have the equation y=e3x+2…. (i) 

If we differentiate with respect to x, we get the following:

y1=dy/dx=e3x+2⋅ d/dx(3x+2)

⇒y1=e3x+2(3)

⇒y1=3e3x+2…(ii)

When we differentiate with respect to x, we get the following:

y2=d/dx(dy/dx)=d2ydx2=3[e3x+2]⋅d/dx(3x+2)

⇒y2=3⋅(e3x+2)⋅(3)

⇒y2=9e3x+2

∴y2=9y.

Conclusion

Successive differentiation is the process of differentiating a given function consecutively times, and the outcomes of such differentiation are referred to as successive derivatives. It is possible to calculate the derivative of a function of a real variable in mathematics by measuring the sensitivity of the function value (output value) to changes in its argument. A good example is the derivative of the position of a moving object with respect to time, which is also known as the object’s velocity. 

Differentiation is the term used to describe the process of determining a derivative. Antidifferentiation is the term used to describe the opposite process. Successive differentiation is the process of deriving higher-order derivatives of a function by differentiating the function successively. In scientific and engineering applications, higher-order differential coefficients are of critical importance because they allow for more accurate calculations. This process, which is known as successive differentiation, can be defined to occur n times. Differentiating a given function several times is known as successive differentiation.