derivative question involving finding dy/dx?

Question

hi i would like to find out how to solve this problem

i am asking again as in my previous question a few days ago that i posted neithe of the answers were correct out of the two responses

i would like to find if y=(x+5)^4/(2x-5)^7 i would like to find dy/dx

Thanks for any help. it is much appreciated

Answer 1

y=(x+5)^4/(2x-5)^7

So dy/dx = [4(x+5)^3 (2x-5)^7 - (x+5)^4 . 7(2x-5)^6 . 2] / (2x-5)^14
= (x+5)^3 (2x-5)^6 (4(2x-5) - (x+5)(14)) / (2x-5)^14
= (x+5)^3 (8x - 20 - 14x - 70) / (2x-5)^8
= (x+5)^3 (-6x - 90) / (2x-5)^8
= -6 (x + 15) (x + 5)^3 / (2x - 5)^8.

Answer 2

Use the quotient rule and the chain rule OR if you prefer, logarithmic differentiation

y=(x+5)^4/(2x-5)^7 ===> then log each side
ln y = ln [ (x+5)^4/(2x-5)^7 ] ===> use log rules to simplify
ln y = 4 ln (x+5) - 7 ln(2x-5) ===> now do derivative each side
1/y * y ' = 4 / (x+5) - 7*2 / (2x-5) ====> solve for y'
y ' = y * [ 4/(x+5) - 14/(2x-5) ] ===> now plug in original y
y ' = (x+5)^4/(2x-5)^7 [ 4/(x+5) - 14/(2x-5) ]

Answer 3

http://en.wikipedia.org/wiki/Quotient_rule

g(x)=(x+5)^4; h(x)=(2x-5)^7

[g'(x)*h(x) - g(x)*h'(x)] / h(x)^2

4*(x-5)*(2x-5)^7 - (x+5)^4*7*(2x-5)^6*2] / (2x-5)^14

[4*(x-5)*(2x-5)^7 - 14*(x+5)^4*(2x-5)^6] / (2x-5)^14