Question

We are given that
dy/dx=√ (1 − y ²). find d²y/dx².

Answer 1

To find d²y/dx², we need to take the derivative of dy/dx with respect to x. Applying the chain rule, we differentiate both sides of the equation dy/dx = √(1 - y²) and obtain d²y/dx² = (d(√(1 - y²))/dy) * (√(1 - y²)).

Step-by-step explanation:

To find d²y/dx², we need to take the derivative of dy/dx with respect to x. Since dy/dx = √(1 - y²), we can differentiate both sides of this equation using the chain rule. Let's start by differentiating the right side:

d(dy/dx)/dx = d(√(1 - y²))/dx

To simplify this, we can convert it into a form where we only have y's and dy's. Therefore, we need to express dx in terms of dy:

dx = 1/dx

Now, let's differentiate the right side:

d(dy/dx)/dx = d(√(1 - y²))/dx

Using the chain rule, we have:

(d(√(1 - y²))/dy) * (dy/dx)

Substituting the expression for dy/dx from the original equation:

(d(√(1 - y²))/dy) * (√(1 - y²))

Therefore, d²y/dx² = (d(√(1 - y²))/dy) * (√(1 - y²)).