Question

Find the differential \(d y\) of the given function. $$ y=\sqrt{9-x^{2}} $$

Short answer

The differential \( dy \) of the function \( y = \sqrt{9-x^2} \) is \( dy= -x / \sqrt{9-x^2} * dx\) .

Step-by-step solution

Rewrite the function

The function \(y = \sqrt{9-x^2}\) is a square root function, which is difficult to differentiate directly. Instead, rewrite the function using fractional exponents to make differentiation easier: \(y = (9-x^2)^{1/2}\)

Differentiate using the Chain Rule

The Chain Rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. In this case, the outer function is \(u^{1/2}\) and the inner function is \(9 - x^{2}\). So, \(dy/dx = 1/2 (9-x^2)^{-1/2} * (d/dx(9 - x^2)) = 1/2 (9-x^2)^{-1/2} * -2x = -x (9-x^2)^{-1/2}.

Simplify the function

Now the function is simplified: \(dy/dx = -x(9-x^2)^{-1/2} = -x / \sqrt{9-x^2}\). Then, by identifying \( dy = dy/dx * dx \) , the differential dy is finally found to be \( dy= -x / \sqrt{9-x^2} * dx\) .