y=cos^-1[(1-x^2)/(1+x^2)]
find dy/dx
To find the derivative of y with respect to x, or dy/dx, we can use the chain rule and the derivative of the inverse cosine function.
Let's break down the problem step by step:
1. Start with the given equation:
y = cos^(-1)((1 - x^2)/(1 + x^2))
2. Rewrite the equation using the inverse cosine function:
cos(y) = (1 - x^2) / (1 + x^2)
3. Differentiate both sides of the equation implicitly with respect to x:
-sin(y) * dy/dx = (d/dx)((1 - x^2) / (1 + x^2))
4. Simplify the right side using the quotient rule:
-sin(y) * dy/dx = [(1 + x^2) * (-2x) - (1 - x^2) * (2x)] / (1 + x^2)^2
5. Simplify further:
-sin(y) * dy/dx = [-2x - 2x^3 + 2x - 2x^3] / (1 + x^2)^2
-sin(y) * dy/dx = -4x^3 / (1 + x^2)^2
6. Divide both sides by -sin(y):
dy/dx = -4x^3 / [(1 + x^2)^2 * -sin(y)]
Now we have the derivative of y with respect to x, dy/dx, in terms of x and y. Keep in mind that this is a implicit derivative, so it's still in terms of x and y.