Use logarithmic differentiation to find the derivative of the function : y = [(x - 1) / (x^4 + 1]^1/2

Question

Use logarithmic differentiation to find the derivative of the function :

 

y = √[(x - 1) / (x⁴ + 1)]

 

 

ANSWER: √[(x - 1) / (x⁴ + 1)]  *  [(1 / (2x - 2)  -  2x³ / (x⁴ + 1)]

#41, p223

Answer 1

If this is the function:

 

y = (x-1)^1/2/(x⁴+1), then the given answer is incorrect (the -2x³ should be -4x³).

 

Take the natural log of both sides and then apply properties of logarithms to rewrite the right side of the equation.

 

ln y = 1/2 ln (x-1) - ln(x⁴+1)

 

Now take the derivative of both sides with respect to x.

 

(1/y) y' = (1/2)[1/(x-1)] - [1/(x⁴+1)](4x³)   :The 4x³ made its appearance based on using the chain rule.

 

Now multiply both sides by y to isolate y'.

 

But y = the mess on the right side of the original problem.

 

So,

 

y' = (x-1)^1/2/(x⁴+1) {(1/2)[1/(x-1)] - [1/(x⁴+1)](4x³) }

 

it is possible to change the way this looks to:

 

y' = (-7x⁴+8x³+1)/2(x-1)^1/2(x⁴+1)²

 

You can get the same result by using the quotient rule.

 

Checkout:

 

http://www.wolframalpha.com 

 

Enter differentiate (function) to confirm.