Respuesta :

JeanaShupp

Answer: [tex]\dfrac{2e^{x^\frac{1}{2}}}{\sqrt{x}}[/tex] .

Step-by-step explanation:

Properties of derivative :

  • [tex]\dfrac{d}{dx}(e^x)=e^x[/tex]  
  • [tex]\dfrac{d}{dx}(x^n)=x^{n-1}[/tex]   (1)
  • [tex]\dfrac{d}{dx}(ax)=a[/tex]
  • Let g be a differentiable function , then [tex]\dfrac{d}{dx}e^g=e^g\dfrac{dg}{dx}[/tex]   (2)

Given function : [tex]y = 4e^{x^{\frac{1}{2}}}[/tex]

Now , differentiate both sides of the above equation with respect to x , we get

[tex]y'=4e^{x^\frac{1}{2}}\dfrac{d(x^{\frac{1}{2}})}{dx}[/tex]  (Using(2))

[tex]\Rightarrow\ y'=4e^{x^\frac{1}{2}}(\dfrac{1}{2}x^{(\frac{1}{2}-1}))[/tex]   (Using  (1))

[tex]\Rightarrow\ y'=2e^{x^\frac{1}{2}}x^{-\frac{1}{2}}[/tex]

[tex]\Rightarrow\ y'=\dfrac{2e^{x^\frac{1}{2}}}{\sqrt{x}}[/tex]  

Hence, the derivative of the given function is [tex]\dfrac{2e^{x^\frac{1}{2}}}{\sqrt{x}}[/tex] .

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