Question

find dy for ( y = e^{5sqrt{x}-1} ).
for ( y = e^{5sqrt{x}-1} ), ( dy=) dx
(type an exact answer, using radicals as needed.)

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:

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