choose the correct setup below to start differentiating the function.\na. ( f^{prime}(x)=\frac{d}{d…

choose the correct setup below to start differentiating the function.\na. ( f^{prime}(x)=\frac{d}{d x}left(e^{sqrt{x}}\right)-\frac{d}{d x}left(e^{sqrt{5}}\right) )\nb. ( f^{prime}(x)=\frac{\frac{d}{d x}left(e^{sqrt{x}}\right)}{\frac{d}{d x}left(e^{sqrt{5}}\right)} )\nc. ( f^{prime}(x)=\frac{d}{d x}left(e^{sqrt{x}}\right)-e^{sqrt{5}} )\nd. ( f^{prime}(x)=e^{sqrt{x - 5}} \frac{d}{d x}(sqrt{x - 5}) )\n( f^{prime}(x)=square )

choose the correct setup below to start differentiating the function.\na. ( f^{prime}(x)=\frac{d}{d x}left(e^{sqrt{x}}\right)-\frac{d}{d x}left(e^{sqrt{5}}\right) )\nb. ( f^{prime}(x)=\frac{\frac{d}{d x}left(e^{sqrt{x}}\right)}{\frac{d}{d x}left(e^{sqrt{5}}\right)} )\nc. ( f^{prime}(x)=\frac{d}{d x}left(e^{sqrt{x}}\right)-e^{sqrt{5}} )\nd. ( f^{prime}(x)=e^{sqrt{x - 5}} \frac{d}{d x}(sqrt{x - 5}) )\n( f^{prime}(x)=square )

Answer

Explanation:

Step1: Apply the chain rule

The chain rule states that if (y = e^{u}), then (y^\prime=e^{u}\cdot u^\prime). Here (u = \sqrt{x - 5}=(x - 5)^{\frac{1}{2}}).

Step2: Differentiate (u=(x - 5)^{\frac{1}{2}})

Using the power rule ((x^n)^\prime=nx^{n - 1}), we have (u^\prime=\frac{1}{2}(x - 5)^{\frac{1}{2}-1}\cdot(x - 5)^\prime). Since ((x - 5)^\prime = 1), then (u^\prime=\frac{1}{2\sqrt{x - 5}}).

Step3: Combine using the chain rule

(f^\prime(x)=e^{\sqrt{x - 5}}\cdot\frac{1}{2\sqrt{x - 5}})

Answer:

(f^\prime(x)=\frac{e^{\sqrt{x - 5}}}{2\sqrt{x - 5}})