What is the derivative of this function?

Question

Can someone tell me how to find the derivative of the function x^1/2 (3−2 x^2) over x?

f(x)= x^1/2 (3−2 x^2)
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x

Answer 1

You know that x^(1/2) / x =1/x^(1/2)
so it's easier to derivate:

f(x)=(3-2x^2)/ x^(1/2)

now using (u/v)'=(u'v-uv')/v^2
with u(x)= 3-2x^2 and v(x)=x^(1/2)
it's easy

u'(x)= -4x and v'(x)=1/ 2x^1/2)

thus f'(x)=(-4x*x^(1/2)-(3-2x^2)*(1/2x^(1/2)))/ x

and then f'(x)=(-3)*(2x^2+1) / (x*x^(1/2))

Answer 2

f (x) = [ 3 x^(1/2) - 2 x^(5/2) ] / x
f (x) = 3 x^(-1/2) - 2 x^(3/2)
f `(x) = (- 3/2) x^( - 3/2 ) - 3 x^(1/2)
f `(x) = (- 3/2)x^( - 3/2 ) [1 + 2 x ² ]
f `(x) = (- 3) (1 + 2 x ²) / 2 x^(3/2)

Answer 3

f(x) = x^1/2 (3 - 2x^2)/x
f(x) = (3 - 2x^2)x^-1/2
f'(x) = - (1/2)(3 - 2x^2)/x^3/2 - 4x^1/2
f'(x) = - (3/2 - 3x^2)/x^3/2
f'(x) = 3(x^2 - 1/2)/x^3/2