Question

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

Short answer

The differential \(dy\) is \(-\frac{x dx}{\sqrt{9-x^{2}}}\).

Step-by-step solution

Rewrite the function

The given function is a composite function and it might need a little rewrite before differentiation is performed. Let's rewrite the function as, \(y=(9-x^{2})^{0.5}\)

Apply the Chain Rule

Applying the chain rule, the outer function is \(u^{0.5}\) and the inner function is \(u = 9 - x^{2}\). First differentiate the outer function with respect to \(u\) and then differentiate the inner function with respect to \(x\).\n Derivative of the outer function \(u^{0.5}\) with respect to \(u\) is \(0.5u^{-0.5}\) and derivative of the inner function \(9 - x^{2}\) with respect to \(x\) is \(-2x\).

Write down the differential

Substituting these into the chain rule gives \(\frac{d y}{d x} = 0.5*(9-x^{2})^{-0.5} * (-2x)\). This can be further simplified to \(\frac{d y}{d x} = -\frac{x}{\sqrt{9-x^{2}}}\). Representing dy gives dy = \(\frac{d y}{d x} dx = -\frac{x dx}{\sqrt{9-x^{2}}}\).