Please help!!
Differentiate:
y=(x^2-1)^(1/2)/x^2
Chain and quotient rule
((1/2)(2x)(x^2-1)^-1/2)x^2 -2x(x^2-1)^1/2)/x^4
simplifies to
(x^3(x^2-1)^-1/2 -2x(x^2-1)^1/2)/x^4
simplifies to
(x^2-1)^-1/2(x^2)-2(x^2-1))/x^3
final answer
((x^2-1)^(-1/2))(-x^2+2)/x^3
To differentiate the given function, we can follow these steps:
Step 1: Rewrite the function
y = ((x^2 - 1)^(1/2)) / x^2
Step 2: Expand the numerator using the power rule
y = ((x^2 - 1)^(1/2)) * x^(-2)
Step 3: Apply the quotient rule
The quotient rule states that for a function u(x)/v(x), the derivative is given by:
(u'v - uv') / v^2
In our case, u(x) = (x^2 - 1)^(1/2) and v(x) = x^2. Let's calculate the derivatives of u(x) and v(x).
Derivative of u(x):
To find the derivative of u(x), we need to apply the chain rule since we have a composite function. The chain rule states that if u(x) = f(g(x)), then u'(x) = f'(g(x)) * g'(x).
Let f(u) = u^(1/2), and g(x) = x^2 - 1.
Using the power rule, f'(u) is given by (1/2) * u^(-1/2).
Taking the derivative of g(x) with respect to x, we get g'(x) = 2x.
Now, let's apply the chain rule:
u'(x) = (1/2) * ((x^2 - 1)^(1/2))^(-1/2) * 2x
= x / ((x^2 - 1)^(1/2))
Derivative of v(x):
To find the derivative of v(x) = x^2, we can use the power rule.
Taking the derivative of v(x) with respect to x, we get v'(x) = 2x.
Step 4: Apply the quotient rule formula
Using the formula (u'v - uv') / v^2, we can substitute the derivatives we calculated:
y' = ((x / ((x^2 - 1)^(1/2))) * x^2 - ((x^2 - 1)^(1/2)) * 2x) / (x^2)^2
= (x^3 / ((x^2 - 1)^(1/2))) - 2x((x^2 - 1)^(1/2))) / x^4
= x^3 / ((x^2 - 1)^(1/2) * x^4) - 2((x^2 - 1)^(1/2)) / x^3
Therefore, the derivative of y = (x^2 - 1)^(1/2) / x^2 is:
y' = x^3 / ((x^2 - 1)^(1/2) * x^4) - 2((x^2 - 1)^(1/2)) / x^3