find dy for ( y = e^{5sqrt{x}-1} ).\nfor ( y = e^{5sqrt{x}-1} ), ( dy=) dx\n(type an exact answer, using…

Explanation:

Step1: Let (u = 5\sqrt{x}-1)

Then (y = e^{u}).

Step2: Differentiate (u) with respect to (x)

Using the power rule (\frac{d}{dx}(x^{n})=nx^{n - 1}), (\frac{du}{dx}=\frac{5}{2\sqrt{x}}).

Step3: Differentiate (y) with respect to (u)

Since (\frac{dy}{du}=e^{u}).

Step4: Use the chain - rule (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx})

Substitute (u = 5\sqrt{x}-1), (\frac{dy}{dx}=e^{5\sqrt{x}-1}\cdot\frac{5}{2\sqrt{x}}).

Step5: Find (dy)

Since (dy=\frac{dy}{dx}dx), so (dy=\frac{5e^{5\sqrt{x}-1}}{2\sqrt{x}}dx).

Answer:

(\frac{5e^{5\sqrt{x}-1}}{2\sqrt{x}}dx)