Find dy/dx for `y = (3x - 1)*sqrt x`

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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The function `y=(3x - 1)*sqrt x`

`y = 3*x*sqrt x - sqrt x`

=> `y = 3*x^(3/2) - sqrt x`

dy/dx = `(3/2)*3*x^(3/2 - 1) - (1/2)*x^(1/2 - 1)`

=> `dy/dx = (9/2)*sqrt x - 1/(2*sqrt x)`

The required value of `dy/dx = (9/2)*sqrt x - 1/(2*sqrt x)`

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You may also use the product rule of such that:

dy/dx = (3x - 1)'*sqrt x + (3x - 1)*(sqrt x)'

dy/dx = 3sqrt x + 3x/(2sqrt x) - 1/(2 sqrt x)

dy/dx = 3sqrt x + 3(sqrt x)^2/(2sqrt x) - 1/(2 sqrt x)

Bringing all fractions top a common denominator yields:

dy/dx = (6x + 3x - 1)/(2sqrt x)

Adding like powers of x yields:

dy/dx = (9x - 1)/(2sqrt x)

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dstell77 | Student, Undergraduate | eNotes Newbie

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Thank so much, you are so helpful!
!!

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