find the derivative of f when f(x)= (4x^3/2 + x^-3/2)^2?

Question

A) 4(16+x^-6) / (x^3)
B) 3(4x^3+1) / (x^3)
C) 3(16x^6 -1) / (x^4)
D) 4( 16 - x^-6) / (x^3)
E) 3(4x^6-1) / (x^3)

Answer 1

f(x) = [4x^(3/2) + x^(-3/2)]^2
f'(x) = 2[4x^(3/2) + x^(-3/2)][(4*3/2)x^(3/2 - 1) + (-3/2)x^(-3/2 - 1)]
f'(x) = 2[4x^(3/2) + x^(-3/2)][6x^(1/2) - (3/2)x^(-5/2)]
f'(x) = [4x^(3/2) + x^(-3/2)][12x^(1/2) - 3x^(-5/2)]
f'(x) = [(4x^3 + 1) / x^(3/2)] [(12x^3 - 3) / x^(5/2)]
f'(x) = [(4x^3 + 1)(12x^3 - 3)] / [x^(3/2)*x^(5/2)]
f'(x) = [(4x^3 + 1)(12x^3 - 3)] / (x^4)
f'(x) = 3[(4x^3 + 1)(4x^3 - 1)] / (x^4)
f'(x) = 3(16x^6 -1) / (x^4)

So (C) is correct.

Answer 2

f (x) = 16.x³ + 8 + x^(-3)
f `(x) = 48x² - 3.x^(-4)
f `(x) = 48x² - 3 / (x^4)
f `(x) = 3.(16x^6 - 1) / (x^4)
ANSWER C)