Question

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

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:

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