Question

Differentiate implicitly to find the first partial derivatives of \(w\). \(w-\sqrt{x-y}-\sqrt{y-z}=0\)

Short answer

The first partial derivatives of \(w\) are \(\frac{dw}{dx} = \frac{1}{2\sqrt{x-y}}\), \(\frac{dw}{dy} = -\frac{1}{2\sqrt{x-y}} + \frac{1}{2\sqrt{y-z}}\), and \(\frac{dw}{dz} = -\frac{1}{2\sqrt{y-z}}\).

Step-by-step solution

Differentiating w.r.t. x

Differentiate the given expression with respect to \(x\). Using the chain rule and the fact that the derivative of \(\sqrt{x}\) with respect to \(x\) is \(1/(2\sqrt{x})\), we have \(\frac{dw}{dx} = \frac{1}{2\sqrt{x-y}}\).

Differentiating w.r.t. y

Next, differentiate the given expression with respect to \(y\). Again using the chain rule and the fact that the derivative of \(\sqrt{x}\) with respect to \(x\) is \(1/(2\sqrt{x})\), we have \(\frac{dw}{dy} = -\frac{1}{2\sqrt{x-y}} + \frac{1}{2\sqrt{y-z}}\).

Differentiating w.r.t. z

Finally, differentiate the given expression with respect to \(z\). Using the chain rule we again have \(\frac{dw}{dz} = -\frac{1}{2\sqrt{y-z}}\).