Question
find the first and second derivative of x/(x^2-2)
Answers
GPT 3.5
To find the first derivative of the given function, we can use the quotient rule.
Let f(x) = x/(x^2 - 2).
Using the quotient rule, the first derivative f'(x) is given by:
f'(x) = [(x^2 - 2)(1) - (x)(2x)] / (x^2 - 2)^2
Simplifying this expression, we get:
f'(x) = (x^2 - 2 - 2x^2) / (x^2 - 2)^2
= (-x^2 - 2) / (x^2 - 2)^2
To find the second derivative, we differentiate f'(x) with respect to x. Let's call the second derivative g(x):
g(x) = (-x^2 - 2) / (x^2 - 2)^2
Using the quotient rule again, g'(x) is given by:
g'(x) = [((x^2 - 2)^2)(-2x) - (-x^2 - 2)(2(x^2 - 2)(2x))] / (x^2 - 2)^4
Simplifying this expression, we have:
g'(x) = (-2x(x^2 - 2)^2 + 4x(x^2 - 2)(2x)(x^2 + 2)) / (x^2 - 2)^4
= (-2x(x^2 - 2)^2 + 4x(x^2 - 2)(x^2 + 2)(2x)) / (x^2 - 2)^4
= (-2x(x^2 - 2)^2 + 4x(x^2 - 2)(x^4 + 2x^2)(2x)) / (x^2 - 2)^4
Therefore, the first derivative is f'(x) = (-x^2 - 2) / (x^2 - 2)^2, and the second derivative is g'(x) = (-2x(x^2 - 2)^2 + 4x(x^2 - 2)(x^4 + 2x^2)(2x)) / (x^2 - 2)^4.
Let f(x) = x/(x^2 - 2).
Using the quotient rule, the first derivative f'(x) is given by:
f'(x) = [(x^2 - 2)(1) - (x)(2x)] / (x^2 - 2)^2
Simplifying this expression, we get:
f'(x) = (x^2 - 2 - 2x^2) / (x^2 - 2)^2
= (-x^2 - 2) / (x^2 - 2)^2
To find the second derivative, we differentiate f'(x) with respect to x. Let's call the second derivative g(x):
g(x) = (-x^2 - 2) / (x^2 - 2)^2
Using the quotient rule again, g'(x) is given by:
g'(x) = [((x^2 - 2)^2)(-2x) - (-x^2 - 2)(2(x^2 - 2)(2x))] / (x^2 - 2)^4
Simplifying this expression, we have:
g'(x) = (-2x(x^2 - 2)^2 + 4x(x^2 - 2)(2x)(x^2 + 2)) / (x^2 - 2)^4
= (-2x(x^2 - 2)^2 + 4x(x^2 - 2)(x^2 + 2)(2x)) / (x^2 - 2)^4
= (-2x(x^2 - 2)^2 + 4x(x^2 - 2)(x^4 + 2x^2)(2x)) / (x^2 - 2)^4
Therefore, the first derivative is f'(x) = (-x^2 - 2) / (x^2 - 2)^2, and the second derivative is g'(x) = (-2x(x^2 - 2)^2 + 4x(x^2 - 2)(x^4 + 2x^2)(2x)) / (x^2 - 2)^4.