find dy for ( y = e^{sqrt{x}} - 2 ).\nfor ( y = e^{sqrt{x}} - 2 ), ( dy = ) dx.\n(type an exact answer…

Explanation:

Step1: Differentiate the function using the chain rule

Let ( u = \sqrt{x}-2=x^{\frac{1}{2}} - 2). Then ( y = e^{u}). The derivative of (y) with respect to (u) is (\frac{dy}{du}=e^{u}), and the derivative of (u) with respect to (x) is (\frac{du}{dx}=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}). By the chain rule (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). Substituting (u = \sqrt{x}-2) into (\frac{dy}{du}), we get (\frac{dy}{du}=e^{\sqrt{x}-2}). So (\frac{dy}{dx}=e^{\sqrt{x}-2}\cdot\frac{1}{2\sqrt{x}}).

Step2: Find (dy)

Since (dy=\frac{dy}{dx}dx), substituting (\frac{dy}{dx}=e^{\sqrt{x}-2}\cdot\frac{1}{2\sqrt{x}}) into (dy=\frac{dy}{dx}dx), we have (dy = \frac{e^{\sqrt{x}-2}}{2\sqrt{x}}dx).

Answer:

(\frac{e^{\sqrt{x}-2}}{2\sqrt{x}})