derivative!!!!?

Question

ive been working on this impossible webquiz all afternoon and i just finished...now i have to do derivatives...
brain cramp...
what is the derivative of
f(x)=(4x^8-sqrt(x))
-------------------
(8x^4)
im trying to simplify and its coming out really complicated!!!!

ooh shoot! I didnt see that you could cancle 8x^4!!!!!
that makes it so much easier..i multiplied it out and got this huge..messy answer!!
thanks michael!!

Answer 1

Break it apart into two fractions first, then take the derivative.

f(x)=(4x^8-sqrt(x)) / (8x^4)
f(x)=(x^4)/2 - 1 / (8x^(7/2)
f(x)=(1/2)x^4 - (1/8)x^(-7/2)

f'(x) = 4*(1/2)x^(4-1) - (1/8)(-7/2)x^(-7/2-1)
f'(x) = 2x^3 + (7/16)x^(-9/2)
f'(x) = 2x^3 + 7/[16x^(9/2)]

Answer 2

Assuming you mean (4x^8 - sqrt(x)) / (8x^4)
Use the quotient rule
derv[(4x^8 - sqrt(x)) / (8x^4)]
={ (8x^4)*derv[4x^8 - sqrt(x)] - (4x^8 - sqrt(x))*derv[8x^4] } / (8x^4)^2
={ (8x^4)*[32x^7 - (1/2)(x)^(-1/2)] - (4x^8 - sqrt(x))*32x^3 } / (8x^4)^2
={ [32x^7 - (1/2)(x)^(-1/2)] - (4x^7 - (x)^(-1/2))*4 } / (8x^4)
={ [64x^7 - (x)^(-1/2)] - (32x^7 - 8(x)^(-1/2)) } / (16x^4)
={ 64x^7 - (x)^(-1/2) - 32x^7 + 8(x)^(-1/2) } / (16x^4)
={ 32x^7 + 7(x)^(-1/2) } / (16x^4)
=(32x^8 + 7sqrt(x)) / (16x^5)
=32x^8 / (16x^5) + 7sqrt(x) / (16x^5)
=2x^3 + 7 / (16x^(4.5))

Yes these things are long; do more of them to build confidence.

edit: yes northstar's way is faster and simpler, and you get the same answer

Answer 3

f(x) = (4x^8 - sqrt(x))

f'(x) = 4*8*x^7 - 1/2 x^(-1/2)

If the second part is f(x) = 8x^4 then f'(x) = 8*4*x^3