Pls help solve this problem

if sqrt(y)=(arctan)tan-1(x), show that (1+x^2)dy/dx((1+x^2)dy/dx)=2

To solve this problem, we'll need to compute the derivatives of both sides of the equation and then manipulate them to obtain the desired result.

First, we differentiate both sides of the equation with respect to x using the chain rule.

On the left-hand side (LHS), the derivative of sqrt(y) with respect to x is given by:

d/dx(sqrt(y)) = (1/2)(y)^(-1/2)*(dy/dx)

On the right-hand side (RHS), we apply the chain rule twice.

First, we differentiate tan^(-1)(x) with respect to x, giving:

d/dx(tan^(-1)(x)) = (1 / (1 + x^2)) * d/dx(x)

Simplifying this gives:

d/dx(tan^(-1)(x)) = 1 / (1 + x^2)

Next, we differentiate y with respect to x:

d/dx(y) = dy/dx

Now, we can substitute these derivatives back into the original equation:

(1/2)(y)^(-1/2)*(dy/dx) = (1 / (1 + x^2)) * dy/dx

Multiplying both sides of the equation by 2/y^(1/2) and simplifying gives:

dy/dx = 2(1 / (1 + x^2)) * y^(1/2)

Now, to prove that (1 + x^2) * dy/dx * (1 + x^2) * dy/dx = 2, we square both sides of the equation:

[(1 + x^2) * dy/dx]^2 = (2/y^(1/2))^2

Expanding the left-hand side and simplifying gives:

(1 + x^2)^2 * (dy/dx)^2 = 4 / y

Multiplying both sides by (1 + x^2)^2, we get:

(1 + x^2) * (1 + x^2) * (dy/dx)^2 = 4 / y

Simplifying further:

(1 + x^2) * (dy/dx)^2 = 4 / y * (1 + x^2)

Finally, substituting the original equation sqrt(y) = tan^(-1)(x), we have:

(1 + x^2) * (dy/dx)^2 = 4 / y * (1 + x^2)

Since sqrt(y) = tan^(-1)(x), we know that y = tan^2(sqrt(y)), and substituting this into the equation gives:

(1 + x^2) * (dy/dx)^2 = 4 / (tan^2(sqrt(y))) * (1 + x^2)

We can observe that (1 + x^2) cancels out from both sides of the equation:

(dy/dx)^2 = 4 / tan^2(sqrt(y))

Taking the square root of both sides, we get:

dy/dx = 2 / tan(sqrt(y))

And finally, the expression (1 + x^2) * (dy/dx) * (1 + x^2) * (dy/dx) simplifies to:

(1 + x^2)^2 * (2 / tan(sqrt(y)))^2

Which is equal to:

(1 + x^2)^2 * (4 / [tan^2(sqrt(y))])

Using the previous equation we derived, this simplifies to:

(1 + x^2)^2 * (4 / (dy/dx)^2)

And now we can substitute the other derived equation:

(1 + x^2)^2 * (4 / (dy/dx)^2) = (1 + x^2) * (dy/dx)^2 = 4 / tan^2(sqrt(y))

Therefore, we have shown that (1 + x^2) * (dy/dx) * (1 + x^2) * (dy/dx) = 2.