How to Find Derivative of Root x
There are two methods to find the derivative of root x:
- Using First Principle of Differentiation
- Using Power Rule
Derivative of Root x Using First Principle
First principle of differentiation state that derivative of a function f(x) is defined as,
f'(x) = limhβ0 [f(x + h) β f(x)]/[(x + h) β x]
f'(x) = limhβ0 [f(x + h) β f(x)]/ h
Putting f(x) = βx, to find derivative of root x, we get,
f'(x) = limhβ0 [β(x + h) β β(x)]/ h
Multiplying numerator and denominator by β(x + h) + β(x), we get,
β limhβ0 [β(x + h) β β(x)]Γ[β(x + h) + β(x)]/[hΓ(β(x + h) + β(x))]
β limhβ0 [|x+h-x|] / [hΓ(β(x + h) + β(x))]
β limhβ0 [h] / [hΓ(β(x + h) + β(x))]
β limhβ0 1/ [(β(x + h) + β(x))]
β 1/ [(β(x + 0) + β(x))]
β 1/2βx
Hence, we have derived the derivative of root x by using first principle of differentiation.
Derivative of Root x Using Power Rule
Root x is an algebraic function which can be represented as x1/2. The Power Rule in differentiation states that,
For any function of the form xn, where n is any real number, the derivative of the function is nxn-1.
Applying the power rule to find derivative of x1/2, we get,
(x1/2)β = 1/2(x)1/2-1
β 1/2(x)-1/2
β 1/2x1/2 or 1/2βx
Thus, we derived the derivative of root x using the Power Rule.
Derivative of Root x
Derivative of Root x is (1/2)x-1/2 or 1/(2βx). In general, the derivative of a function is defined as the change in the dependent variable, i.e. y = f(x) with respect to the independent variable, i.e. x. This process, also known as differentiation in calculus. Root x is an abbreviation used for the square root function which is mathematically represented as βx or x1/2 (x raised to the power half).
In this article, we will discuss the derivative in math, the derivative of root x, various methods to derive it including the first principle method and the power rule, some solved examples, and practice problems.
Contact Us