write the derivative of f(x) = `sqrtx``` ``

2 Answers

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to convert the radical into a rational power, such that:

`sqrt x = x^(1/2)`

You need to use the following formula of differentiation, such that:

`(x^n)' = n*x^(n - 1)`

Reasoning by analogy, yields:

`(x^(1/2))' = (1/2)*x^(1/2 - 1) => (x^(1/2))' = (1/2)*x^(-1/2)`

Converting the negative power into a positive power yields:

`(x^(1/2))' = (1/2)*1/(x^(1/2))`

Converting back the rational power into radical yields:

`(sqrt x)' = 1/(2sqrt x)`

Hence, evaluating the derivative of the function `y = sqrt x` , using the indicated formula, yields `y' = (sqrt x)' = 1/(2sqrt x).`

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pramodpandey | College Teacher | (Level 3) Valedictorian

Posted on

`y=sqrt(x)`       (i)

squaring both side

`y^2=x`       (ii)

differentiate  (ii) with respect to x ,we have

`2y(dy)/(dx)=1` ``      

`(dy)/(dx)=1/(2y)`             (iii)

substitute y from (i) in (iii) ,we have