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: Let \( u = 5\sqrt{x}+3\)

Then \( y = e^{u}\). By the chain rule \( dy/dx=\frac{dy}{du}\cdot\frac{du}{dx}\).
First, find \(\frac{dy}{du}\):
Since \( y = e^{u}\), \(\frac{dy}{du}=e^{u}\).
Second, find \(\frac{du}{dx}\):
Given \( u = 5x^{\frac{1}{2}}+3\), using the power rule \(\frac{d}{dx}(ax^{n})=anx^{n - 1}\), we have \(\frac{du}{dx}=5\times\frac{1}{2}x^{\frac{1}{2}-1}+0=\frac{5}{2\sqrt{x}}\).

Step2: Apply the chain - rule

By the chain rule \( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\). Substitute \(\frac{dy}{du}=e^{u}\) and \(\frac{du}{dx}=\frac{5}{2\sqrt{x}}\) into the formula.
Since \( u = 5\sqrt{x}+3\), \(\frac{dy}{dx}=e^{5\sqrt{x}+3}\cdot\frac{5}{2\sqrt{x}}\).
And \( dy=\frac{5e^{5\sqrt{x}+3}}{2\sqrt{x}}dx\).

Answer:

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