What is the first derivative of function given by f(x)=(x-square rootx)/(5x^2-3) ?

We have to find the derivative of (x - sqrt x) / (5x^2 -
3).

As we have the expression as a quotient, we use the
quotient rule.

f(x) = u(x) /
v(x)

=> f'(x) = (u'(x)*v(x) -
u(x)*v'(x))/v(x)^2

Here, u(x) = x - sqrt x, v(x) = 5x^2 -
3

u'(x) = 1 - (1/2)(1/sqrt x) = 1 - 1/(2*sqrt x) = (2*sqrt
x - 1)/(2*sqrt x)

v'(x) =
10x

f'(x) = [(5x^2 - 3)(2*sqrt x - 1)/(2*sqrt x) - (x -
sqrt x)*10x]/(5x^2 - 3)^2

f'(x) = [(5x^2 - 3)(2*sqrt x - 1)
- 10x*(x - sqrt x)*(2*sqrt x)] / (2*sqrt x)*(5x^2 -
3)^2

f'(x) = [sqrt x*(10x^2 - 6) - 5x^2 + 3 - 20x^2*sqrt x
+ 20x^2] / (2*sqrt x)*(5x^2 - 3)^2

f'(x) = [-sqrt x*(10x^2
+ 6) + 15x^2 + 3] / (2*sqrt x)*(5x^2 - 3)^2

f'(x) = - [sqrt
x*(10x^2 + 6) - 15x^2 - 3] / (2*sqrt x)*(5x^2 -
3)^2

The required derivative is [-sqrt
x*(10x^2 + 6) + 15x^2 + 3] / [(2*sqrt x)*(5x^2 -
3)^2]