Rational numbers on number line Rational numbers on number line In calculus power rule is one of the most important rule that is defined for differentiating the linear or polynomial terms. It is denoted as the differentiation of the terms d (x n) / d x = n x n - 1. So it states that if there is a function that has the power ‘n’ then at the time of differentiation term is multiplied with the power and power is subtracted by one. Here ‘n’ is any number as we can define it by some simple examples: If the value of n = 3 then f (x) = x3 is differentiate as d (f (x)) / d x = d x3 / d x = 3 x (3 – 1) = 3 x 2. If the value of n = 6 then f (x) = x6 is differentiate as d (f (x)) / d x = d x6 / d x = 3 x (6 – 1) = 3 x 5. If the value of n = 1 then f (x) = x1 is differentiate as d (f (x)) / d x = d x1 / d x = 3 x (1 – 1) = 3 x0 = 3 *1 = 3.
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Here some examples that have the power in negative numbers then. If the value of n = -3 then f (x) = x -3 is differentiate as d (f (x)) / d x = d x-3 / d x = -3 x (-3 – 1) = -3 x -4. We also calculate the derivative if the expression f (x) = 1 / x as we can write the expression 1 / x = x -1 then according to the power rule d f (x) / d x = d x -1 / d x = -1 x (-1 – 1) = -1 x -2. If the value of ‘n’ that is power of an expression also be in fraction as if the expression is f (x) = x 1 / 2 then it is also written as f (x) = √ x is differentiated as, d f (x) / d x = d √ x / d x = d x 1 / 2 / d x = 1 / 2 (x ) 1 / 2 – 1 = 1 / 2 (x) -1 / 2. So in power rule power can be anything not necessarily a whole number it may be a number or fraction or ∏. If we talk about the inverse of the power rule that is defined in the integration have all the powers except n = -1 is defined as ∫ y n d y = (y n + 1 / n + 1) + c where n > = 0. Here ‘c’ is a constant. It is also known as the power rule for integration that also has the linear or polynomial expressions. That is also explained by some examples as if the expression f (a) = a 99 then ∫ a 99 d a = (a 99 + 1 / 99 + 1) + c = (a 100 / 100) + c .
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Numbers appear like dancing letters to many students as they are not able to distinguish between different categories of numbers and get confused in understanding their concepts. Number family has numerous of siblings and one of them is irrational and Rational Numbers. You can define rational number as a nameable number, as we can name it in the whole numbers, fractions and mixed numbers. On the other side irrational number is one that can’t be expressed in simple fraction form. With the help of real life examples you can easily distinguish between differen
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