Rates of change can be found with the help of the differentiation. We can find the rate of change of many factors such as the velocity of some object with respect to the time. Thus, in simple words, with the help of differentiation we can find the change ratio of one object with respect to the corresponding change that is happening in another object. It can be seen on a graph consists of y and x and the x is the curve’s gradient. The derivative of a certain non-linear function with some value of variable x, which is independent, can be identified with the help of differentiation or the delta method. This particular working method consists of a algebraic approach and includes a lot of work which is quite hard. There are many rules which are in fact very basic that allow us to differentiate any non-linear function without any difficulty hence these rules are very easy.
Before we start taking about the rules, we have to know to about the terms and notations that we use regularly while working with the differentiation. These terms and notations are important to know as they very confusing to the students. When students start working with the differentiation they find it very difficult only if they are not clearly familiar with all the terms and the notations. The calculus has the use of many notations and all of them are in some different forms however, all the notations are used to represent or address the same thing or result. There are four types of notations, which are available in the domain of differentiation. The used notations are Leibniz's notation, Newton's notation, Lagrange's notation and Euler's notation.
Leibniz's notation: The famous mathematician Gottfried Wilhelm Leibniz introduce this method in the year 1675. The symbols that he used are dx, dy and dx/dy. This is the most common method while defining the functional relationship between the independent and dependent variables such as y = f(x). The first form of derivative looks like dy/dx and df/dx. The higher form of the derivatives looks like dny/dxn and dnf/dxn. When we do the derivative for the nth time of the equation y=f(x), the previous notations are used. The second order derivative loos like d2y/dx2 = d/dx (dy/dx). The Leibniz's notation make the rule of chain very essy to remember and the rule is dy/dx = dy/du * du/dx.
Newton's notation: The notations that the Newton used is also often called as the dot notation. The main feature of this kind of notation is that there is dot placed at the top of the every function name for representing the time derivative. For the function y=f(t) the functions can be written as the y and the top of the y, there should be dot placed for the first order derivative and at the top of the y there should be two dots placed to represent the second order derivative. When we have to find out the derivative of some function with respect to arc length or time we use the notations denoted by Newton. In the domain physics while solving the problems related to differential equations we commonly use the Newton’s notations. The use of the Newton’s notations also has the use in the domain of differential geometry. In case of high order derivatives, these notations have no use as they cannot denote or deal with the independent variables with the multiple or higher order.
Lagrange's notation: These notations are often represented as the prime notation. In modern days these notations are the most common notations to use. The mathematician Joseph-Louis Lagrange first introduced these notations. The main feature of this notation is the use of prime mark. Thus, the derivative of function f is represented by the term f′ and this is the first order derivative of the function f. The notation can be used to denote the second and third order derivative. f′′ and f′′′ is used to denote the second and the third order derivatives respectively. When we have to find the derivatives beyond the third order, many mathematicians use the roman numerals in the form of superscript to represent the order.
Euler's notation: This notation has the use of differential operator D. D is applied to certain functions to figure out the first order derivative of the function such as Df. Dnf denotes the nth order derivative of the function f. Sometime the independent variable is attached to the operator D to clarify the difference between independent and dependent variable such as Dxy or Dxf(x).
The following section is the detailed discussion about the various different rules for differentiation. These rules are all different while remembering or applying. There are five rules to remember and use and they are constant rule, constant multiple rule, power rule, difference rule and sum rule.
The constant rule: This is a very simple rule to use and remember while solving the differential equations. Assume a horizontal line consisting a zero slope, f(x) = 5. Thus, for c which is any number, f(x)=c, then the derivative is f′ (x) = 0 or it can be written as dc/dx = 0.
The power rule: Suppose the equation is given as the f(x)=x5. If we have to the derivative then the power 5 should be taken and brought at the front of x. Then the power have to be reduce by 1 and the power then becomes 4. So now we have f′(x)=5x4. Then to find the result, this step has to be repeated.
The constant multiple rule: If a function begins with the coefficient then we have to ignore the coefficient as they have no effect. Example of such function is y=4x3. Here the we know derivative of function x3 should be 3x2. But as there is 4 placed at start of the function hence we have to multiply 4 with 3 and the result will be y′=12x2.
The sum rule: When we have to find the derivative of terms, which is presented as combination of sum then we have to find the derivatives of the terms separately. The example of such function is f′(x) of f(x) = x6+x3+x2+x+10 is f′(x)=6x5+3x2+2x+1.
The difference rule: This is same as the previous rule with negative signs between each terms. Here also we have to find the derivative of each term separately. Example is y=3x5-x4-2x3+6x2+5x then the required derivative is y′=15x4-4x3-6x2+12x+5. The signs between the terms are unaffected when differentiation is done.
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