Question

find dy for ( y = e^{5sqrt{x}+3} ).
for ( y = e^{5sqrt{x}+3} ), ( dy = ) dx.
(type an exact answer, using radicals as needed.)

Explanation:

Step1: Differentiate the function using the chain rule

The chain rule states that if \(y = e^{u}\), then \(y^\prime=e^{u}\cdot u^\prime\). Let \(u = 5\sqrt{x}+3=5x^{\frac{1}{2}}+3\). First, find \(u^\prime\):
\(u^\prime=\frac{d}{dx}(5x^{\frac{1}{2}}+3)\)
Using the power rule \(\frac{d}{dx}(x^n)=nx^{n - 1}\), we have \(u^\prime=5\times\frac{1}{2}x^{\frac{1}{2}-1}+0=\frac{5}{2}x^{-\frac{1}{2}}=\frac{5}{2\sqrt{x}}\)
Since \(y = e^{u}\), by the chain rule \(y^\prime=\frac{dy}{dx}=e^{u}\cdot u^\prime\)
Substitute \(u = 5\sqrt{x}+3\) and \(u^\prime=\frac{5}{2\sqrt{x}}\) into the formula: \(\frac{dy}{dx}=e^{5\sqrt{x}+3}\cdot\frac{5}{2\sqrt{x}}\)

Step2: Find \(dy\)

We know that \(dy=\frac{dy}{dx}dx\)
Since \(\frac{dy}{dx}=\frac{5e^{5\sqrt{x}+3}}{2\sqrt{x}}\), then \(dy=\frac{5e^{5\sqrt{x}+3}}{2\sqrt{x}}dx\)

Answer:

\(dy=\frac{5e^{5\sqrt{x}+3}}{2\sqrt{x}}dx\)