find dy for ( y = e^{5sqrt{x}+3} ).\nfor ( y = e^{5sqrt{x}+3} ), ( dy = ) dx.\n(type an exact answer, using…

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)