find dy for y = e^{3\\sqrt{x}-2}.\nfor y = e^{3\\sqrt{x}-2}, dy = ( ) dx\n(type an exact answer, using…

Explanation:

Step1: Let ( u = 3\sqrt{x}-2)

First, find the derivative of (u) with respect to (x). Using the power rule, if (u = 3x^{\frac{1}{2}}-2), then (\frac{du}{dx}=\frac{3}{2}x^{-\frac{1}{2}}=\frac{3}{2\sqrt{x}}).

Step2: Use the chain - rule

The function (y = e^{u}). By the chain - rule (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). Since (\frac{dy}{du}=e^{u}) (because the derivative of (y = e^{u}) with respect to (u) is (e^{u})) and (u = 3\sqrt{x}-2), (\frac{dy}{dx}=e^{3\sqrt{x}-2}\cdot\frac{3}{2\sqrt{x}}).

Step3: Find (dy)

Since (dy=\frac{dy}{dx}dx), substituting (\frac{dy}{dx}) we get (dy=\frac{3e^{3\sqrt{x}-2}}{2\sqrt{x}}dx).

Answer:

(\frac{3e^{3\sqrt{x}-2}}{2\sqrt{x}})