What is and how do you find the derivative of The square root of [ (2x^3 -7) / (4x +1)]?

Answer 1

Can you use logs? It may be easier if you can.
Let y = sqrt [ (2x^3 -7) / (4x+1) ] = [ (2x^3 -7) / (4x+1) ]^(1/2)
Take the ln of each side
ln y = ln ([ (2x^3 -7) / (4x+1) ]^(1/2))
Simplify
ln y = (1/2) ln ([ (2x^3 -7) / (4x+1) ])
ln y = (1/2) [ ln(2x^3 -7) - ln (4x+1)]
Now take the derivative of each side.
y' / y = (1/2) [ (6x)/(2x^3 -7) - 4/(4x+1)]
Multiply both sides by y
y' = (1/2) [ (6x)/(2x^3 -7) - 4/(4x+1)] y
Replace y with sqrt [ (2x^3 -7) / (4x+1) ] (from line 2 above)
y' = (1/2) [ (6x)/(2x^3 -7) - 4/(4x+1)] sqrt [ (2x^3 -7) / (4x+1) ]

Answer 2

First put it into this form it makes it easier...

(2x^3-7)^1/2 / (4X+1)

= [1/2(2x^3-7)^(3/2)x 6x^2 x(4x+1) ]- [(2x^3-7)^(1/2) x 4]
all divided by (4x+1)^2


= [3x^2(2x^3-7)^(3/2) x (4x+1)] - [4(2x^3-7)^(1/2)
all divided by (4x+1)^2

That is the derivative....and you can turn it into the old square root form by...when there is an exponent of (3/2) its the third root) and the exponent (1/2) is the square root

Answer 3

6

Answer 4

woooow....I'm sorry, but this will take forever to do, but I can explain it. Change the square root to an exponent of (1/2). Then use the chain rule, which is f '(x) x f(x). To find the derivative of the function, you have to do the quotient rule--for a function (u/v), dervative = (vu'-uv')/v^2. Goooood luck

Answer 5

Use the chain rule.
Let f(x) = (2x^3 -7) / (4x +1).
Differentiating sqrt(f(x)) gives 0.5(f(x))^-0.5 * f'(x).
Thus we need to differentiate f(x).
Probably the easiest way to do that is using the quotient rule, which gives:
((6x^2)(4x+1) - 4(2x^3-7))/(4x+1)^2.

Put f(x) and f'(x) in that original formula, and thats your answer.

Answer 6

first findout thre derivative of [(2x^3-7)/(4x+1)]
then use the chain rule
1/2[(2x^3-7)/(4x+1)]^(-1/2)*derivative of[(2x^3-7)/(4x+1)]

Answer 7

Deep wondering is thinking deep topics. topics that require extreme idea in protecting with different idea to go back to conclusions. Deep topics require continual go back to the concern as your renewed understanding and adulthood promises new perception on the concern. least puzzling answer is continuously best, even in deep wondering. (through the way, your lengthy equation is erroneous. you may want to multiply first, so it is going to change into 4+a million-6, which isn't 9. that is -a million.)

Answer 8

Use the Chain Rule.

Answer 9

use the chain rule

Answer 10

use the chain rule.