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Check my work please, how to simplify this differentiated function?
Differentiate the following function:
y=( x / (x^2 + 1))^2
I did:
y'= 2(x/(x^2+1) ((x^2+1)(1) - (x)(2x) / ((x^2+1)^2)
y' = 2(x / (x^2+1)) ((x^2+1-2x^2) / ((x^2+1)^2)
Now to simplify this, do I cancel out the (x^2+1) in the second term in the parenthesis? Not sure where to go from here to simplify this down..any help would be really appreciated!
1 Answer
- 7 years ago
Simplify the function by multiplying the top and bottom by x^2 + 1. The easy way to differentiate the function is to use log. That is:
ln(y) = 2ln(x/(x^2 + 1))
ln(y) = 2(ln(x) - ln(x^2 + 1))
ln(y) = 2ln(x) - 2ln(x^2 + 1)
Then, by implicit differentiation, we obtain:
1/y * dy/dx = (2/x) - 2 * 2x/(x^2 + 1)
dy/dx = y((2/x) - 4x/(x^2 + 1))
dy/dx = (x/(x^2 + 1))^2 * ((2/x) - 4x/(x^2 + 1))
dy/dx = 2/(x^2 + 1)^2 - 4x^2/(x^2 + 1)^3
Hope this helps!