How do you find the derivative of #(2sqrtx - 1)/(2sqrtx)#?

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Answer 1 · 2017-01-08T18:10:18

#d/(dx) (frac (2sqrt(x)-1) (2sqrt(x)) )= 1/(4sqrt(x^3)#

We could use the quotient rule, stating that:

#d/(dx) ((f(x))/(g(x)) )= (f'(x)g(x)-f(x)g'(x))/(g(x)^2)#

It is however easier to write the function as:

# frac (2sqrt(x)-1) (2sqrt(x)) = 1-1/(2sqrt(x))=1-1/2x^(-1/2)#

so that:

#d/(dx) (frac (2sqrt(x)-1) (2sqrt(x)) )= d/(dx) ( 1-1/2x^(-1/2)) = 1/4x^(-3/2) =1/(4sqrt(x^3)#

Answer 2 · 2017-01-08T18:14:14

Break it into two fractions, the first becomes the derivative of a constant, and the second can be differentiated as a negative power.

Break into two fractions:

#(d((2sqrt(x) - 1)/(2sqrt(x))))/dx = (d((2sqrt(x))/(2sqrt(x))))/dx - (d(1/(2sqrt(x))))/dx#

Simplify:

#(d((2sqrt(x) - 1)/(2sqrt(x))))/dx = (d(1))/dx - 1/2(d(x^(-1/2)))/dx#

The derivative of a constant is zero and use the power rule on the second term:

#(d((2sqrt(x) - 1)/(2sqrt(x))))/dx = 1/4x^(-3/2)#