Question

Find \(d y / d x\) by implicit differentiation. $$ x^{1 / 2}+y^{1 / 2}=9 $$

Short answer

The derivative \(dy/dx\) in the equation \(x^{1 / 2}+y^{1 / 2}=9\) is \(dy/dx = -\sqrt{(9 - \sqrt{x})/x}\).

Step-by-step solution

Re-express the given equation in a form that makes it easier to differentiate

The given equation is \(x^{1/2} + y^{1/2} = 9\). It would be easier to differentiate if we rewrite the \(x^{1/2}\) as \(\sqrt{x}\) and \(y^{1/2}\) as \(\sqrt{y}\). The equation then becomes \(\sqrt{x} + \sqrt{y} = 9\).

Differentiate both sides of the Equation

Once we have rewritten the equation in an easier-to-handle form, we can differentiate both sides with respect to x. For differentiating with respect to x, we use: \(d(\sqrt{x})/dx = 1/(2\sqrt{x})\) and \(d(\sqrt{y})/dx = (1/(2\sqrt{y})) * (dy/dx)\). This is because \(\sqrt{y}\) is with respect to y, so by chain rule, we have to differentiate it with respect to y and then multiply it by \(dy/dx\). After differentiating both sides, we get: \(1/(2\sqrt{x}) + (1/(2\sqrt{y})) * dy/dx = 0\).

Isolate dy/dx

After differentiating we need to isolate dy/dx on one side of the equation to find its value. We do that by subtracting \(1/(2\sqrt{x})\) from both sides and then multiplying both sides by \(2\sqrt{y}\). The equation becomes \( dy/dx = -\sqrt{y/x}\).

Substituting y

Now, we know that from the original equation \(\sqrt{x} + \sqrt{y} = 9\), so \(\sqrt{y} = 9 - \sqrt{x}\). Now we can substitute this back into the equation. So, \( dy/dx = -\sqrt{(9 - \sqrt{x})/x}\).